Patent Publication Number: US-2011055655-A1

Title: Hardware-Efficient Low Density Parity Check Code for Digital Communications

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a Divisional of and claims priority under 35 U.S.C. §120 to U.S. application Ser. No. 11/463,236, filed on Aug. 8, 2006; which is a Continuation of U.S. application Ser. No. 10/329,597, filed on Dec. 26, 2002—now U.S. Pat. No. 7,178,080. Said Continuation claims priority, under 35 U.S.C. §119(e), to Provisional Application No. 60/403,668, filed Aug. 15, 2002. All said applications incorporated herein by reference. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not applicable. 
     BACKGROUND OF THE INVENTION 
     This invention is in the field of digital data communications, and is more specifically directed to redundant coding for error detection and correction in such communications. 
     High-speed data communications, for example in providing high-speed Internet access, is now a widespread utility for many businesses, schools, and homes. At this stage of development, such access is provided according to an array of technologies. Data communications are carried out over existing telephone lines, with relatively slow data rates provided by voice band modems (e.g., according to the current v.92 communications standards), and at higher data rates using Digital Subscriber Line (DSL) technology. Another modern data communications approach involves the use of cable modems communicating over coaxial cable, such as provided in connection with cable television services. The Integrated Services Digital Network (ISDN) is a system of digital phone connections over which data is transmitted simultaneously across the world using end-to-end digital connectivity. Localized wireless network connectivity according to the IEEE 802.11 standard has become very popular for connecting computer workstations and portable computers to a local area network (LAN), and often through the LAN to the Internet. Wireless data communication in the Wide Area Network (WAN) context, which provides cellular-type connectivity for portable and handheld computing devices, is expected to also grow in popularity. 
     A problem that is common to all data communications technologies is the likelihood of corruption of data due to noise. As is fundamental in the art, the signal-to-noise ratio for a communications channel is a degree of goodness of the communications carried out over that channel, as it conveys the relative strength of the signal that carries the data (as attenuated over distance and time), to the noise present on that channel. These factors relate directly to the likelihood that a data bit or symbol received over the channel will be in error relative to the data bit or symbol as transmitted. This likelihood is reflected by the error probability for the communications over the channel, commonly expressed as the Bit Error Rate (BER) ratio of errored bits to total bits transmitted. In short, the likelihood of error in data communications must be considered in developing a communications technology. Techniques for detecting and correcting errors in the communicated data must be incorporated for the communications technology to be useful. 
     Error detection and correction techniques are typically implemented through the use of redundant coding of the data. In general, redundant coding inserts data bits into the transmitted data stream that do not add any additional information, but that indicate whether an error is present in the received data stream. More complex codes provide the ability to deduce the true transmitted data from a received data stream, despite the presence of errors. 
     Many types of redundant codes that provide error correction have been developed. One type of code simply repeats the transmission, for example repeating the payload twice, so that the receiver deduces the transmitted data by applying a decoder that determines the majority vote of the three transmissions for each bit. Of course, this simple redundant approach does not necessarily correct every error, but greatly reduces the payload data rate. In this example, a predictable likelihood remains that two of three bits are in error, resulting in an erroneous majority vote despite the useful data rate having been reduced to one-third. More efficient approaches, such as Hamming codes, have been developed toward the goal of reducing the error rate while maximizing the data rate. 
     The well-known Shannon limit provides a theoretical bound on the optimization of decoder error as a function of data rate. The Shannon limit provides a metric against which codes can be compared, both in the absolute and relative to one another. Since the time of the Shannon proof, modern data correction codes have been developed to more closely approach the theoretical limit. An important type of these conventional codes are “turbo” codes, which encode the data stream by applying two convolutional encoders. One convolutional encoder encodes the datastream as given, while the other encodes a pseudo-randomly interleaved version of the data stream. The results from the two encoders are interwoven to produce the output encoded data stream. 
     Another class of known redundant codes is the Low Density Parity Check code. According to this class of codes, a sparse matrix H defines the code, with the encodings t of the payload data satisfying: 
       Ht=0   (1)
 
     over Galois field GF(2). Each encoding t consists of the source message s combined with the corresponding parity check bits for that source message s. The encodings t are transmitted, with the receiving network element receiving a signal vector r=t+n, n being the noise added by the channel. Because the decoder at the receiver knows matrix H, it can compute a vector z=Hr. However, because r=t+n, and because Ht=0: 
         z=Hr=Ht+Hn=Hn    (2)
 
     The decoding process thus involves finding the sparsest vector x that satisfies the equation: 
       Hx=z   (3)
 
     over GF(2). The vector x becomes the best guess for noise vector n, which can be subtracted from the received signal vector r to recover encodings t, from which the original source message s is recoverable. There have been many examples of LDPC codes that are known in the art, and these LDPC codes have been described as providing code performance that approaches the Shannon limit, as described in Tanner et al., “A Class of Group-Structured LDPC Codes”,  ISTCA- 2001  Proc . (Ambleside, England, 2001). 
     In general, high-performance LDPC code decoders are difficult to implement into hardware. In contrast to Shannon&#39;s adage that random codes are good codes, it is regularity that allows efficient hardware implementation. To address this difficult tradeoff between code irregularity and hardware efficiency, the technique of belief propagation provides an iterative implementation of LDPC decoding can be made somewhat efficient, as described in Richardson, et al., “Design of Capacity-Approaching Irregular Low-Density Parity Check Codes,”  IEEE Trans. on Information Theory , Vol. 47, No. 2 (February 2001), pp. 619-637; and in Zhang et al., “VLSI Implementation-Oriented (3,k)-Regular Low-Density Parity-Check Codes”,  IEEE Workshop on Signal Processing Systems  (September 2001), pp. 25.-36. Belief propagation decoding algorithms are also referred to in the art as probability propagation algorithms, message passing algorithms, and as sum-product algorithms. 
     In summary, belief propagation algorithms are based on the binary parity check property of LDPC codes. As mentioned above and as known in the art, each check vertex in the LDPC code constrains its neighboring variables to form a word of even parity. In other words, the product of the LDPC code word vector with each row of the parity check matrix sums to zero. According to the belief propagation approach, the received data are used to represent the input probabilities at each input node (also referred to as a “bit node”) of a bipartite graph having input nodes and check nodes. Within each iteration of the belief propagation method, bit probability messages are passed from the input nodes to the check nodes, updated according to the parity check constraint, with the updated values sent back to and summed at the input nodes. The summed inputs are formed into log likelihood ratios (LLRs) defined as: 
     
       
         
           
             
               
                 
                   
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     where c is a coded bit received over the channel. 
     In its conventional implementation, the belief propagation algorithm uses two value arrays, a first array L(q mj ) storing the LLRs for the input nodes, and the second array R mj  storing the results of the parity check node updates, with m being the parity check row index and j being the column (or input node) index. The general operation of this conventional approach determines, in a first step, the R mj  values by estimating, for each check sum (row of the parity check matrix) the probability of the input node value from the other inputs used in that checksum. The second step of this algorithm determines the LLR L (q mj ) probability values by combining, for each column, the R mj  values for that input node from parity check matrix rows in which that input node participated. A “hard” decision is then made from the resulting probability values, and is applied to the parity check matrix. This two-step iterative approach is repeated until the parity check matrix is satisfied (all parity check rows equal zero, GF(2)), or until another convergence criteria is reached, or a terminal number of iterations have been executed. 
     By way of further background, the code design approach described in Boutillon et al., “Decoder-First Code Design”,  Proc.: Int&#39;l Symp. on Turbo Codes and Related Topics  (Brest, France, September 2001) defines the decoder architecture first, and uses this architecture to constrain the design of the LDPC code itself. Sridhara, et al., “Low Density Parity Check Codes from Permutation Matrices”, 2001  Conference on Information Sciences and Systems  (Johns Hopkins University, Mar. 21-23, 2001) describes the LDPC code as constructed from shifted identity matrices (i.e., permutation matrices). 
     However, it has been observed in connection with this invention, that these prior approaches are somewhat limited, in that these approaches are limited to a single code or a small selection of codes. Practically useful communications receivers require some amount of flexibility in code rates, and in optimizing their operation for varying noise levels and channel conditions. 
     BRIEF SUMMARY OF THE INVENTION 
     It is therefore an object of this invention to provide an LDPC decoding scheme which can be efficiently implemented in an integrated circuit. 
     It is a further object of this invention to provide such a scheme that is flexible over a wide range of code rates. 
     It is a further object of this invention to provide such a scheme having the capability of parallelism, to provide further efficiencies in operation and construction. 
     Other objects and advantages of this invention will be apparent to those of ordinary skill in the art having reference to the following specification together with its drawings. 
     The present invention may be implemented in connection with a network receiver, having a decoder that implements a Low-Density Parity-Check (LDPC) code for retrieving the transmitted message. The LDPC code is implemented according to a parity check matrix consisting of an irregular arrangement of cyclically shifted identity matrices, resulting in an irregular LDPC code that provides performance near the Shannon limit. A decoder architecture for this code includes a group of column sum memories that receive the received input data, and that accumulate and store updated values for the input node predictions. A reversible router block forwards these column, input node, values to a parity check update block, at which multiple predictions are generated for each input node, one prediction for each parity check (row) in which the input node is involved; a prediction memory is also provided for storing these predictions. The outputs of the parity check update block are forwarded through the router, and accumulated in the column sum memories. 
     According to another aspect of the invention, the invention is implemented by encoding a datastream by applying a systematic block code corresponding to an irregular arrangement of circularly shifted identity matrices. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
         FIG. 1  is a functional block diagram of communications between two OFDM transceivers, where at least the receiving transceiver is constructed according to a first preferred embodiment of the invention. 
         FIG. 2  is an electrical diagram, in block form, of a transceiver constructed according to the preferred embodiments of the invention. 
         FIG. 3  is a flow chart illustrating a method of designing an LDPC code according to the preferred embodiments of the invention. 
         FIGS. 4   a  and  4   b  are examples of LDPC code macro matrices according to the preferred embodiments of the invention. 
         FIG. 5  is an electrical diagram, in block form, of an LDPC decoder according to a first preferred embodiment of the invention. 
         FIG. 6  is an electrical diagram, in block form, of a parity check and update circuit in the LDPC decoder of  FIG. 5 , according to the first preferred embodiment of the invention. 
         FIG. 7  is an electrical diagram, in block form, of an example of routing circuitry in the LDPC decoder of  FIG. 5 , according to the first preferred embodiment of the invention. 
         FIG. 8  is an electrical diagram, in block form, of a bit update circuit in the LDPC decoder of  FIG. 5 , according to the first preferred embodiment of the invention. 
         FIG. 9  is an electrical diagram, in block form, of an LDPC decoder according to a second preferred embodiment of the invention. 
         FIG. 10  is a timing diagram, illustrating data word misalignment according to the second embodiment of the invention. 
         FIG. 11  is a flow chart illustrating a method for solving data word misalignment according to the second embodiment of the invention. 
         FIGS. 12 and 12   a  are electrical diagrams, in block form, of the construction of a parity check and update circuit according to an alternative embodiment of the invention. 
         FIG. 13  is an electrical diagram, in block form, of the construction of a parallel parity check and update circuit according to another alternative embodiment of the invention. 
         FIGS. 14   a  through  14   g  are electrical diagrams, in block form, of various alternative logical arrangements of memory according to the preferred embodiments of the invention and the physical circuitry for effecting these logical arrangements. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention will be described in connection with an example of its implementation in an exemplary transceiver, for example a wireless network adapter such as according to the IEEE 802.11 wireless standard. It will be apparent to those skilled in the art having reference to this specification that this invention is particularly well-suited for use in such an application. However, it is also contemplated that this invention will be of similar benefit in many other applications that involve error correction coding, including communications according to orthogonal frequency division multiplexing (OFDM), discrete multitone modulation (DMT) for example as used in conventional Digital Subscriber Line (DSL) modems, and other modulation and communication approaches, whether carried out as land line or wireless communications. It is therefore to be understood that these and other alternatives to and variations on the embodiment described below are contemplated to be within the scope of the invention as claimed. 
       FIG. 1  functionally illustrates an example of a somewhat generalized communication system into which the preferred embodiment of the invention is implemented. The illustrated system corresponds to an OFDM modulation arrangement, as useful in OFDM wireless communications as contemplated for IEEE 802.11 wireless networking. The data flow in this approach is also analogous to Discrete Multitone modulation (DMT) as used in conventional DSL communications, as known in the art. It is contemplated that this generalized arrangement is provided by way of context only. In the system of  FIG. 1 , only one direction of transmission (from transmitting transceiver  10  over transmission channel C to receiving transceiver  20 ) is illustrated. It will of course be understood by those skilled in the art that data will also be communicated in the opposite direction, in which case transceiver  20  will be the transmitting transceiver and transceiver  10  the receiving transceiver. 
     As shown in  FIG. 1 , transmitting transceiver  10  receives an input bitstream that is to be transmitted to receiving tranceiver  20 . The input bitstream may be generated by a computer at the same location (e.g., the central office) as transmitting tranceiver  10 , or alternatively and more likely is generated by a computer network, in the Internet sense, that is coupled to transmitting tranceiver  10 . Typically, this input bitstream is a serial stream of binary digits, in the appropriate format as produced by the data source. 
     The input bitstream is received by LDPC encoder function  11 , according to this embodiment of the invention. LDPC encoder function  11  digitally encodes the input bitstream by applying a redundant code for error detection and correction purposes. According to this embodiment of the invention, the redundant LDPC code applied by encoder function  11  is selected in a manner that facilitates implementation and performance of the corresponding decoder in receiving tranceiver  20 . The specifics of the code will become apparent from the description of this decoder function, presented below relative to the description of the construction and operation of receiving tranceiver  20 . In general, the coded bits include both the payload data bits and also code bits that are selected, based on the payload bits, so that the application of the codeword (payload plus code bits) to the sparse LDPC parity check matrix equals zero for each parity check row. After application of the LDPC code, bit to symbol encoder function  11  groups the incoming bits into symbols having a size, for example, ranging up to as many as fifteen bits. These symbols will modulate the various subchannels in the OFDM broadband transmission. 
     The encoded symbols are then applied to inverse Discrete Fourier Transform (IDFT) function  14 . IDFT function  14  associates each input symbol with one subchannel in the transmission frequency band, and generates a corresponding number of time domain symbol samples according to the Fourier transform. These time domain symbol samples are then converted into a serial stream of samples by parallel-to-serial converter  16 . Functions  11  through  16  thus convert the input bitstream into a serial sequence of symbol values representative of the sum of a number of modulated subchannel carrier frequencies, the modulation indicative of the various data values, and including the appropriate redundant code bits for error correction. Typically, for an input of N/2 complex symbols, IDFT function  14  outputs a block of N real-valued time domain samples. Those skilled in the art having reference to this specification will readily recognize that each of functions  11  through  16  may be carried out, and preferably actually are carried out, as digital operations executed by a digital signal processor (DSP). 
     Filtering and conversion function  18  then processes the datastream for transmission. Function  18  applies the appropriate digital filtering operations, such as interpolation to increase sample rate and digital low pass filter for removing image components, for the transmission. The digitally-filtered datastream signal is then converted into the analog domain and the appropriate analog filtering is then applied to the output analog signal, prior to its transmission. 
     The output of filter and conversion function  18  is then applied to transmission channel C, for forwarding to receiving tranceiver  20 . The transmission channel C will of course depend upon the type of communications being carried out. In the wireless communications context, the channel will be the particular environment through which the wireless transmission takes place. Alternatively, in the DSL context, the transmission channel is physically realized by conventional twisted-pair wire. In any case, transmission channel C adds significant distortion and noise to the transmitted analog signal, which can be characterized in the form of a channel impulse response. 
     This transmitted signal is received by receiving tranceiver  20 , which, in general, reverses the processes of transmitting transceiver  10  to recover the information of the input bitstream. 
       FIG. 2  illustrates an exemplary construction of receiving tranceiver  20 , in the form of a wireless network adapter. Transceiver  20  is coupled to host system  30  by way of a corresponding bus B. Host system  30  corresponds to a personal computer, a laptop computer, or any sort of computing device capable of wireless networking in the context of a wireless LAN; of course, the particulars of host system  30  will vary with the particular application. In the example of  FIG. 2 , transceiver  20  may correspond to a built-in wireless adapter that is physically realized within its corresponding host system  30 , to an adapter card installable within host system  30 , or to an external card or adapter coupled to host computer  30 . The particular protocol and physical arrangement of bus B will, of course, depend upon the form factor and specific realization of tranceiver  20 . Examples of suitable buses for bus B include PCI, MiniPCI, USB, CardBus, and the like. 
     Transceiver  20  in this example includes spread spectrum processor  31 , which is bidirectionally coupled to bus B on one side, and to radio frequency (RF) circuitry  33  on its other side. RF circuitry  33 , which may be realized by conventional RF circuitry known in the art, performs the analog demodulation, amplification, and filtering of RF signals received over the wireless channel and the analog modulation, amplification, and filtering of RF signals to be transmitted by transceiver  20  over the wireless channel, both via antenna A. The architecture of spread spectrum processor  31  into which this embodiment of the invention can be implemented follows that of the TNETW1100 single-chip WLAN medium access controller (MAC) available from Texas 
     Instruments Incorporated. This exemplary architecture includes embedded central processing unit (CPU)  36 , for example realized as a reduced instruction set (RISC) processor, for managing high level control functions within spread-spectrum processor  31 . For example, embedded CPU  36  manages host interface  34  to directly support the appropriate physical interface to bus B and host system  30 . Local RAM  32  is available to embedded CPU  36  and other functions in spread spectrum processor  31  for code execution and data buffering. Medium access controller (MAC)  37  and baseband processor  39  are also implemented within spread-spectrum processor  31  according to the preferred embodiments of the invention, for generating the appropriate packets for wireless communication, and providing encryption, decryption, and wired equivalent privacy (WEP) functionality. Program memory  35  is provided within tranceiver  20 , for example in the form of electrically erasable/programmable read-only memory (EEPROM), to store the sequences of operating instructions executable by spread-spectrum processor  31 , including the coding and decoding sequences according to the preferred embodiments of the invention, which will be described in further detail below. Also included within wireless adapter  20  are other typical support circuitry and functions that are not shown, but that are useful in connection with the particular operation of tranceiver  20 . 
     According to the preferred embodiments of the invention, LDPC decoding is embodied in specific custom architecture hardware associated with baseband processor  39 , and shown as LDPC decoder circuitry  38  in  FIG. 2 . LDPC decoder circuitry  38  is custom circuitry for performing the coding and decoding of transmitted and received data packets according to the preferred embodiments of the invention. Examples of the particular construction of LDPC decoder circuitry  38  according to the preferred embodiment of this invention will be described in further detail below. 
     Alternatively, it is contemplated baseband processor  39  itself, or other computational devices within tranceiver  20 , may have sufficient computational capacity and performance to implement the decoding functions described below in software, specifically by executing a sequence of program instructions. It is contemplated that those skilled in the art having reference to this specification will be readily able to construct such a software approach, for those implementations in which the processing resources are capable of timely performing such decoding. 
     Referring back to the functional flow of  FIG. 1 , filtering and conversion function  21  in receiving transceiver  20  processes the signal that is received over transmission channel C. Function  21  applies the appropriate analog filtering, analog-to-digital conversion, and digital filtering to the received signals, again depending upon the technology of the communications. In the DSL context, this filtering can also include the application of a time domain equalizer (TEQ) to effectively shorten the length of the impulse response of the transmission channel H. Serial-to-parallel converter  23  converts the filtered datastream into a number of samples that are applied to Discrete Fourier Transform (DFT) function  24 . Because, in this OFDM context, the received signal is a time-domain superposition of the modulated subchannels, DFT function  24  recovers the modulating symbols at each of the subchannel frequencies, reversing the IDFT performed by function  14  in transmitting tranceiver  10 . DFT function  24  outputs a frequency domain representation of a block of transmitted symbols, multiplied by the frequency-domain response of the effective transmission channel. Recovery function  25  then effectively divides out the frequency-domain response of the effective channel, for example by the application of a frequency domain equalizer (FEQ), to recover an estimate of the modulating symbols. Symbol-to-bit decoder function  26  then demaps the recovered symbols, and applies the resulting bits to LDPC decoder function  28 . 
     LDPC decoder function  28  reverses the encoding that was applied in the transmission of the signal, to recover an output bitstream that corresponds to the input bitstream upon which the transmission was based. This output bitstream is then forwarded to the host workstation or other recipient. 
     LDPC Decoding 
     The theory of operation of the preferred embodiment of the invention will now be described, following which its implementation into LDPC decoding function  28  in tranceiver  20 , in the form of LDPC decoder circuitry  38  operating in cooperation with baseband processor  39 , will then be described. 
     By way of nomenclature, the LDPC code is fundamentally contained within an m×j parity check matrix H pc  that, when multiplied by the true transmitted code word vector c equals zero: 
         H   pc   ·c= 0   (5)
 
     over Galois Field (2). For a single one of the m rows in parity check matrix H pc , this parity check amounts to: 
         H   1   c   1   +H   2   c   2   + . . . +H   j   c   j =0   (6a)
 
     over GF(2). In the LDPC code according to the preferred embodiments of the invention, the parity check matrix H pc  is formed from a composite of circularly shifted identity matrices represented by a macro matrix H. Each entry in macro matrix H represents a permutation matrix (e.g., a circularly shifted identity matrix), and in this example takes either a 1 or a 0 value. As will be described below, an entry with a 1 value in macro matrix 
     H symbolizes a p×p permutation matrix at that position within parity check H pc , while entries with a 0 value symbolize a p×p zero matrix. The parity-check equation thus logically becomes, for an exemplary row of matrix H pc  having a “1” in its columns 1, 3, 4, and 7: 
         c   1   ⊕c   3   ⊕c   4   ⊕c   7 =0   (6b)
 
     Once the coding matrix H pc  is defined, the encoding of a message frame is relatively straightforward, as known in the art, and can easily be performed by conventional programmable integrated circuits such as digital signal processors and the like. According to the preferred embodiments of the invention, the circularly shifted identity matrices are tiled within macro matrix H in an irregular manner, as will be described below, to provide excellent coding performance. 
     On the decoding side, one can define a set N(m) as the set of all bit indices (columns) in a given row m for which codeword bits contribute to the checksum (i.e., all bit indices for which the entries of parity check matrix H pc  in row m are 1). The checksum equation for a row of the parity check can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
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     over GF(2) or, logically, the exclusive-OR of the input bits c j  that correspond to column bits in the row having a 1 value. One can thus determine, for a given codeword vector c, whether an error is present by determining whether this equation is true for each row of the parity check matrix H pc . 
     In practice, however, the actual input bit values r j  that are recovered after demodulation and that are to be interpreted as codeword vector c by a decoder, for example by decoding function  28  in transceiver  20  of  FIG. 1 , are not binary values. Rather, these bit values are expressed as a fractional value, for example between zero and one, expressed in several bits (e.g., six or seven). In effect, the input bit values r j  can be considered as, and converted to, probabilities that their respective bit is a 0 (or conversely a 1). As known in this art, the log likelihood ratio (LLR) is a commonly used representation for these probabilities: 
     
       
         
           
             
               
                 
                   
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     which can of course take negative and positive values, corresponding to 1 and 0 being more likely, respectively. For this description of the preferred embodiment of the invention, one can assume that the incoming LLRs (i.e., the received data) have the form −2r j /σ 2  where σ 2  represents channel noise variance. 
     Fundamentally, the LDPC decoding process according to the preferred embodiments of the invention involves an iterative two-step process:
         1. Estimate a value R mj  for each of the j input nodes, for each of the m rows of the checksum, using the current probability values from the other input nodes, setting the result of the checksum for the row to 0; and   2. Update the sum L(q j ) for each of the j input nodes from a combination of the m values of R mj  in the same column.       

     The iterations continue until a termination criterion is reached. A preferred termination criteria is the earlier of (i) evaluation of the matrix operation H pc ·c=0 (mod 2), using “hard” decisions from the LLRs L(r j ) as the codeword vector c, and (ii) completion of a specified number of iterations. 
     Mathematically, for the first step of estimating values R mj  for each of the j input nodes, for each of the m rows of the checksum, one can derive an amplitude A mj  and a sign value s mj  as follows: 
     
       
         
           
             
               
                 
                   
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     where the function Ψ is defined as: 
       Ψ( x )≡log(|tan  h ( x/ 2)|)=log(tan  h|x/ 2|)   (10)
 
     The function Ψ is its own negative inverse: Ψ(Ψ(x))=−|x|. For computational ease, one can express tan h(L/2) as: 
     
       
         
           
             
               
                 
                   
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     The sign is determined from: 
     
       
         
           
             
               
                 
                   
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     which is simply an odd/even determination of the number of negative probabilities, excluding each row&#39;s own contribution. The updated estimate of values R mj  is thus: 
         R   mj   =−s   mj Ψ( A   mj )   (13)
 
     The negative sign of value R mj  contemplates that the function Ψ is its own negative inverse. The value R mj  thus corresponds to an estimate of the LLR for input node j as derived from the other input nodes in the mth row of the parity check matrix, except input node j itself. 
     An alternative computation of the estimate values R mj  can be defined as a special summation: 
     
       
         
           
             
               
                 
                   
                     R 
                     mj 
                   
                   = 
                   
                     
                       ∑ 
                       
                         
                           n 
                           ∈ 
                           
                             N 
                              
                             
                               ( 
                               m 
                               ) 
                             
                           
                         
                         ; 
                         
                           n 
                           ≠ 
                           j 
                         
                       
                     
                      
                     
                       
                         [ 
                         + 
                         ] 
                       
                        
                       
                         L 
                          
                         
                           ( 
                           
                             q 
                             mn 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     where the LLR addition [+] is defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         L 
                          
                         
                           ( 
                           
                             q 
                             u 
                           
                           ) 
                         
                       
                        
                       
                         [ 
                         + 
                         ] 
                       
                     
                      
                     
                       L 
                        
                       
                         ( 
                         
                           q 
                           v 
                         
                         ) 
                       
                     
                   
                   ≡ 
                   
                     log 
                     ( 
                     
                       
                         1 
                         + 
                         
                            
                           
                             ( 
                             
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     q 
                                     u 
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 L 
                                  
                                 
                                   ( 
                                   
                                     q 
                                     v 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                            
                           
                             L 
                              
                             
                               ( 
                               
                                 q 
                                 u 
                               
                               ) 
                             
                           
                         
                         + 
                         
                            
                           
                             L 
                              
                             
                               ( 
                               
                                 q 
                                 v 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     This alternative determination of the estimate values R mj  may be easier to implement into some integrated circuit architectures. The selection of the computations may be made by those skilled in the art having reference to this specification, and confronted by a particular technology and decoding application. 
     In the second step of each decoding iteration, the LLR estimates for each input node are updated. For each column (i.e., each input node): 
     
       
         
           
             
               
                 
                   
                     L 
                      
                     
                       ( 
                       
                         q 
                         j 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           m 
                           ∈ 
                           
                             M 
                              
                             
                               ( 
                               j 
                               ) 
                             
                           
                         
                       
                        
                       
                         R 
                         mj 
                       
                     
                     + 
                     
                       ( 
                       
                         
                           
                             - 
                             2 
                           
                            
                           
                             r 
                             j 
                           
                         
                         
                           σ 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     where the set M(j) is the set of all check sum indices (rows) for a given column j of check sum equations to which input bit j contributes to the checksum (i.e., all row indices for which the entries of parity check matrix H pc  in column j are 1). This operation effectively sums the estimated values R mj  over the jth column, and adds in the original received input node value 
     
       
         
           
             
               
                 - 
                 2 
               
                
               
                 r 
                 j 
               
             
             
               σ 
               2 
             
           
         
       
     
     to form the best full estimate of the LLR for input node j in this iteration.
 
This column estimate will be used in the hard decision check. In preparation for the next iteration, the per-row (or extrinsic) LLR probabilities are then derived:
 
         L ( q   mj )= L ( q   j )− R   mj    (17)
 
     for each column j in each row m. The per-row probabilities thus amount to an estimate for the probability of the input value, excluding the contribution to the estimate for each row from the row itself. 
     As noted above, the determination of whether the iterations have converged to an error free decoding is based on the per-column LLRs L(q j ): 
         L ( q   j )≧0 ,       c   j =0   (18a)
 
         L ( q   j )&lt;0 ,       c   j =1   (18b)
 
     The codeword vector c={c 0 , c 1 , . . . c N } is then applied to the parity check matrix H to determine if the product is zero, which as noted above indicates that codeword vector c is a valid codeword, from which the payload data portion can be readily extracted. 
     In practice, for those value arrays R mj , L(q mj ), A mj  that are used in the algorithm, the computations performed and the non-zero array values occur only at those index positions (m,j) in parity check matrix H pc  , where a “1” value appears. Also in practice, the initialization of the array L(q mj ) can be arbitrary. For example, each of the L(q mj ) values can be initialized to zero, or to the input values −2r j /σ 2 , as desired. 
     According to the preferred embodiment of the invention, an LDPC code is used for encoding data to be transmitted which has the benefits of performing near the Shannon limit, while being implementable in a very efficient manner. The encoding function according to this preferred embodiment of the invention will be described in detail below. However, it is the decoding operation that requires significant computational power, considering that the received datastream is expected to include some frequency of errors, as evident by the use of redundant coding to begin with. Accordingly, the computational complexity and performance of decoding circuitry has significant practical importance, and is often the deciding factor in whether a particular code is practically useful. It will therefore be useful, for purposes of this description, to first describe the code in terms of the decoding algorithm and architecture, prior to describing the encoding function itself, following a description of the code derivation in general terms. 
       FIG. 3  illustrates a method of deriving an LDPC code according to the preferred embodiments of the invention. It is contemplated that the particular construction of the LDPC codes according to this invention will become apparent from the manner in which these codes are constructed. In process  40 , the code rate is selected. This code rate selection of course depends upon the usual factors involved in the communications, including the expected noise level on the channel, the bit-error-rate (BER) performance that is desired, and of course the desired data rate. According to the preferred embodiments of the invention, data rates of ½ and ⅓ are contemplated. However, as will become apparent to those skilled in the art having reference to this description, this invention permits a wide range of flexibility in the selection and implementation of the redundant code, and as such a wide range of data rates are contemplated. 
     In process  42 , the largest input node degree, or variable degree, for the code is selected. As known in the art, this maximum input node degree corresponds to the largest number of checksum rows that involves a given input node; for LDPC codes such as used in this embodiment of the invention, this input node degree corresponds to the maximum number of parity checks that any input node contributes to. While larger input node degrees are desirable, for better error correction capability, the input node degree is ultimately constrained by the hardware implementation. In the examples of the preferred embodiments of the invention described in this specification, the largest variable degree is contemplated to be on the order of ten to twenty. 
     In process  44 , the degree distributions within parity check matrix H pc  are optimized. These degree distributions refer to (i) the input node degree distribution, which is the number of check nodes that each input node contributes to, and (ii) the check node degree distribution, which is the number of input nodes that each check node receives. Regular LDPC codes are those for which all nodes of the same type have the same degree. According to the preferred embodiment of the invention, however, irregular LDPC codes are used, to improve error rate performance, and as such each code will have a distribution of degrees over its input nodes, and possibly also over its check nodes. A preferred example of an optimization tool for process  46  is a web-based tool known as “ldpcopt”, which is readily available at http://lthcwww.epfl.ch/research/ldpcopt/ and is described in Richardson et al., “Design of Capacity-Approaching Irregular Low-Density Parity-Check Codes”,  Transactions on Information Theory , Vol. 47, No. 2 (IEEE, February 2001), pp. 619-637; and in Chung, et al., “Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation”,  Transactions on Information Theory , Vol. 47, No. 2 (IEEE, February 2001), pp. 657-670; both incorporated herein by this reference. The optimization criteria used by this exemplary process minimizes the bit error rate for the code, by varying the variable degrees of the LDPC parity check matrix. The results of the optimization provide the optimum numbers of “1” values (but not position) in parity check matrix H pc , in the form of optimum numbers of “blocks” within its defining macro matrix H. 
     According to the preferred embodiments of the invention, as briefly mentioned above, the parity check matrix H pc  is determined through the use of a macro matrix H. Macro matrix H is a matrix, of dimensions c by d, in which each entry (“block”) indicates the position of a p×p matrix in parity check matrix H pc , and in which the value of each entry symbolizes the form of the corresponding p×p matrix at that position. As a result, parity check matrix H pc  has a total number of rows m=p×c and a total number of columns j=p×d. Each row of macro matrix H thus defines a “block row” of p rows in parity check matrix H pc ; conversely, each column of macro matrix H defines a “block column” of p columns in parity check matrix H pc . As such, this description will refer to rows of macro matrix H as “block rows”, and columns of macro matrix H as “block columns”. In the preferred embodiments of this invention, a zero value of an entry in macro matrix H symbolizes a zero-valued p×p matrix (i.e., all entries in the p×p matrix are zero). A non-zero value (e.g., 1) of an entry in macro matrix H pc  symbolizes the location of a permutation matrix, which in the preferred embodiments of the invention is a cyclically (or circularly) shifted identity matrix. 
     The optimization of process  44  determines, for a given macro matrix H of c block rows and d block columns, the distribution of the “1” values within the matrix. Specifically, the distribution determines the number of block columns of macro matrix H that have each of a range of numbers of “1” values, and the number of block rows of macro matrix H that have each of a range of numbers of “1” values. By way of definition, the block rows and block columns refer to the illustration of macro matrix H that indicates the locations of the permutation matrices by “1” values. Once the optimization of process  44  is performed, the specific size of c block rows and d block columns is selected in process  46  as shown in  FIG. 3 . Alternatively, one may select the numbers of c block rows and d block columns prior to optimization process  44 . In either case, the optimization of process  44  can be illustrated by way of examples. 
     A first example has the following parameter values for a code rate of ½: 
                                             c   d   p   a   b                  12   24   193   7   49                    
In this example, the optimization of process  44 , performed by way of the “ldpcopt” program, provides the following distributions of column blocks:
 
     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                 g 
                 2 per column 
                 3 per column 
                 4 per column 
                 11 per column 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Optimum 
                 10.80 
                 8.90 
                 0.74 
                 3.56 
               
               
                 Useful 
                 11 
                 9 
                 1 
                 3 
               
               
                   
               
            
           
         
       
     
     In this table, the value g is the number of “1” bits in a given block column. As shown in this example, the optimization indicates that the optimum distribution of its twenty-four block columns (d=24) would have 10.80 block columns with two “1” bits, 8.9 block columns with three “1” bits, 0.74 block columns with four “1” bits, and 3.56 block columns with eleven “1” bits. Of course, this arrangement is not possible for a single macro matrix H having discrete rows and columns. As such, the “Useful” values in this table have rounded the optimum value to integer values. 
     Optimization process  44  also provides an optimized allocation of row blocks, by providing an optimized selection of the number of block rows that have varying numbers of “1” values. In this same first example, the optimization process provided the following results: 
                                                 g   7 per row   8 per row                                                        Optimum   5.56   6.64           Useful   10   2                        
As evident from this table, the optimum distribution provided for 5.56 block rows with seven “1” bits in each row, and 6.64 block rows with eight “1” bits in each row, for a total of c=12 rows. The wide variation for the “Useful” allocation from the optimum is due to the number of filled “1” blocks defined for the columns, which is incompatible with the number of filled blocks defined by the optimized allocation of block rows. In this example, the “Useful” column allocation establishes that there are eighty-six filled block matrix entries; the only combination of row allocations that satisfies that optimized column allocation, while maintaining either seven or eight filled “1” blocks in each row, is the “Useful” allocation of ten block rows with seven “1” bits each, and two block rows with eight “1” bits each, as shown in this table. Alternatively, one could arrange the matrix to satisfy the optimum row allocation and adjust the column allocation.  FIG. 4   a  illustrates a macro matrix H constructed according to the arrangement of this example.
 
     According to a second example, for a code rate of ⅓, the parameters are as follows: 
     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                 c 
                 d 
                 p 
                 a 
                 b 
               
               
                   
               
             
            
               
                 16 
                 24 
                 241 
                 2 
                 44 
               
               
                   
               
            
           
         
       
     
     In this example, the optimization of process  44 , performed by way of the “ldpcopt” program, provides the following distributions of column blocks: 
                                             g   2 per column   3 per column   5 per column   15 per column                                                    Optimum   13.22   5.73   3.17   1.88       Useful   13   6   3   2                    
This optimum distribution of the twenty-four block columns (d=24) is approximated, in this case, by relatively close rounding of the optimum counts to the nearest integer. The row allocation in this example is as follows:
 
                                                 g   5 per row   6 per row                                                        Optimum   8.73   7.27           Useful   9   7                        
In this case, the optimum distribution of the sixteen (c=16) block rows is also closely approximated by rounding to the nearest integers, as shown in the “Useful” row of the table. In this case, the optimized block row and block column allocations are compatible.  FIG. 4   b  illustrates an example of macro matrix H according to this optimization.
 
     In process  46 , if not previously selected prior to optimization process  44 , the multiplicative orders c, d are defined, in which order value c and order value d correspond to the number of block rows and block columns, respectively, in macro matrix H, as described above. 
     In process  48 , the code designer next constructs the particular arrangement of macro matrix H according to the optimization results of process  44 , and according to additional constraints according to this embodiment of the invention. According to the preferred embodiments of the invention, the macro matrix H is arranged in groups of block columns, so that, for any given block row in macro matrix H, at most one block column within each group has a filled “1” entry. This arrangement must take into account the manner in which the constituent matrices are developed. 
       FIGS. 4   a  and  4   b  illustrate examples of macro matrix H, following the two optimization examples described above. Each of  FIGS. 4   a  and  4   b  illustrates macro matrix H representing a matrix of matrices. Each entry of macro matrix H itself represents a square matrix of dimensions p×p. Each empty entry (“0”) of macro matrix H constitutes an empty p×p matrix (all entries are “0”). Each filled entry (“1”) of macro matrix H constitutes a permutation p×p matrix, each formed as an identity matrix with its rows cyclically shifted (modulo p), by an amount P s,t =b s a t , where s, t represent the row and column positions, respectively, of the permutation matrix within macro matrix H, and a, b are the generator values. Referring to  FIG. 3 , these additional code parameters p, a, b, are selected in process  49 . These parameters include prime number p that defines the size of the constituent matrices within macro matrix H, and the generators a, b that are used in the definition of the constituent matrices within macro matrix H. Generator a is selected to have multiplicative order d with respect to prime p, and generator b has multiplicative order c also with respect to prime p. The set of parameters p, c, d, define the size of parity check matrix H pc , with the total number of rows m=p×c and the total number of columns j=p×d, and thus define the appearance of the LDPC code. 
     As noted above, generator a is selected to have multiplicative order d, and generator b has multiplicative order c. In process  50 , these cyclically shifted identity matrices are then generated for each of the filled “1” locations of macro matrix H, to produce parity check matrix H pc . As mentioned above, parity check matrix H pc  thus has m=p×c rows and j=p×d columns, with the “1” values determined by generators a, b. The manner in which the permutation matrices are formed is described in Sridhara, et al., “Low Density Parity Check Codes from Permutation Matrices”, 2001  Conference on Information Sciences and Systems  (Johns Hopkins University, March 21-23, 2001), incorporated herein by this reference. 
     In contrast to the Sridhara approach, however, macro matrix H according to the preferred embodiments of this invention is irregular, in that it is itself relatively sparse, including many zero-valued entries. As will be evident from the following description, the irregularity included by the code designer in process  48 , in combination with grouping of block columns of macro matrix H, provides a high performance LDPC code that can be efficiently implemented in hardware. 
     It is also contemplated, in connection with this invention, that the cyclic shifts of the identity matrices corresponding to the row and column position of macro matrix H need not follow this particular approach. Indeed, the offset P s,t  of the cyclic shift within a particular permutation matrix can be an arbitrary selection. Such arbitrary assignment, if followed, may affect the ultimate dimensions of macro matrix H. 
     The examples of  FIGS. 4   a  and  4   b  illustrate irregular macro matrices H constructed according to this embodiment of the invention. The irregularity and the block construction is evident from these two LDPC code macro matrices H. It is contemplated that those skilled in the art, having reference to this specification, will be readily able to construct macro matrices and the resulting corresponding parity check matrices in this manner, for other code rates and performance optimization levels. 
     Referring now to  FIG. 5 , the construction of LDPC decoder  38  according to a preferred embodiment of the invention will now be described in detail. While it is also contemplated, according to this invention, that the LDPC decoding operations described herein may alternatively be performed as a software routine, for example by baseband processor  39  in the form of a DSP or another programmable logic device executing a sequence of program instructions, the exemplary architecture of  FIG. 5  is especially well-suited to the irregular LDPC codes described above, and provide the important benefit of efficient and rapid performance of the iterative belief propagation decoding described above. 
     As shown in  FIG. 5 , LDPC decoder  38  includes memory  52 , which is preferably a local random access memory (RAM) for storing the R mj  estimates that are derived within each iteration of the belief propagation. These R mj  estimates are packed into memory  52 , so that the column positions within memory  52  do not physically align with the column positions within parity check matrix H pc , to save chip area considering the sparseness of matrix H. R storage memory  52  has an output coupled to one input of parallel adder  54 . Specifically, this output of R storage memory  52  is connected to a negative input of parallel adder  54 , to provide the subtrahend for a subtraction performed by parallel adder  54 . The output of parallel adder  54  is applied to parity check update circuitry  56 . Parity check update circuitry  56  performs the updating of estimates for each of the parity check nodes, as will be described in further detail below. As such, the output of parity check update circuitry  56  is applied to R storage memory  52 , for storage of the updated values; in addition, the output of parity check update circuitry  56  is also applied to router circuitry  58   f , for use in updating the log likelihood ratios (LLRs) of the input nodes. 
     Router circuitry  58   f  is a bank of multiplexers and demultiplexers, as will be described in further detail below, that forwards the appropriate estimate terms R mj  to the corresponding bit update circuits  60 . Bit update circuits  60  are effectively accumulators, by way of which current values of the LLRs of the input nodes are maintained from iteration to iteration. The number of bit update circuits  60  depends upon the maximum number of groups of block columns of macro matrix H. In the particular code; for the code example of  FIG. 4   a , nine bit update circuits  60  will be provided, while for the code example of  FIG. 4   b , seven bit update circuits  60  are necessary. The specific construction of bit update circuits  60  will be described in further detail below. 
     Bit update blocks  60  receive the input node data values, prior to the first iteration of the belief propagation. As mentioned above, the received input node data values are typically based on a multiple bit fractional value, for example expressed in six or seven bits, as produced after demodulation and recovery after frequency domain equalization (e.g., by functions  26  and  27  of  FIG. 1 ). More specifically, because LDPC decoder  38  operates on LLRs, the initial input node data value is preferably expressed as the ratio 
     
       
         
           
             
               
                 
                   - 
                   2 
                 
                  
                 
                   r 
                   j 
                 
               
               
                 σ 
                 2 
               
             
             , 
           
         
       
     
     the value r j  being the received data value. Bit update blocks  60  each forward an output to reverse router circuitry  58   r , which in turn applies the output value to parallel adder  54 , as minuends for the subtraction performed by that circuit. In addition, the outputs of bit update blocks  60  are also applied by reverse router circuitry  58   r  to parity check function  61 , which performs a slicing function on these estimates, and after converting these values to “hard” decisions, determines whether the parity check equation is satisfied by the current estimates for each row of parity check matrix H pc . 
     Referring now to  FIG. 5  in combination with  FIGS. 6 through 8 , the operation of LDPC decoder  38  in performing belief propagation decoding according to the preferred embodiment of the invention will now be described in further detail. The specific construction of constituent circuit blocks and functions within decoder  38  will also be described in connection with this operational description. For the sake of clarity, this description will first be provided with respect to an arbitrary selected iteration in the process. The initialization of the belief propagation operation begins with the storage of values corresonding to ratio 
     
       
         
           
             
               
                 
                   - 
                   2 
                 
                  
                 
                   r 
                   j 
                 
               
               
                 σ 
                 2 
               
             
             , 
           
         
       
     
     the value r j  being the received data value, stored in the appropriate memory locations as the initial estimate of the LLRs for the columns. 
     This description of the operation of LDPC decoder  38 , and the detailed construction of its functional blocks, for this exemplary iteration, will begin at parallel adder  54 . Parallel adder  54  receives the current estimates R i   mj  for the participating columns in a given row m of macro matrix H, from memory  52 . These current estimates R i   mj , which were generated in the previous iteration i of the process, are subtracted from the current LLR values L(q j ) of the input nodes that participate in the current row m of parity check matrix H pc , thus generating the LLR values L(q mj ) according to Equation (17) described above. These values L(q mj ) are forwarded to parity check update circuitry  56 , the construction of which will be described in detail relative to  FIG. 6 . 
     The updating process as carried out by parity check update circuitry  56  begins with the application of each of the values L(q mj ) in the input word received from parallel adder  54  to look-up tables (LUTs)  80 , by way of which the Ψ function of Equation (10) is evaluated for each of the values L(q mj ) within the current row m. The outputs of LUTs  80  are forwarded to augmented adder tree  82 , which performs the summing of the values Ψ(L(q mj )) over all of the columns participating in the current row m of macro matrix H. Augmented adder tree  82  effects this summation in a manner that can be readily implemented by those skilled in the art having reference to this specification. This overall sum result is applied to an input of adders  86 , one adder  86  associated with each of the columns j contributing to the current row m. Each adder  86  also receives, at a negative input, the output of its corresponding LUT  80 , and thus subtracts the column&#39;s own contribution from the overall sum. The outputs of adders  86  thus present the set of amplitude values A mj  corresponding to the result of Equation (9), each associated with one of the columns j that are participating in this row. The outputs of adders  86  are then again applied to corresponding LUTs  88 , to again apply the Ψ function to the amplitude values A mj , according to Equation (13). Sign correction functions  90  apply the appropriate sign to the output of LUTs  88 , based on the logical combination of the sign bit outputs of LUTs  80  for the corresponding column according to an odd/even determination of the number of negative probabilities, excluding each row&#39;s own contribution, as described above relative to Equation (12), and effecting the negative sign applied according to Equation (13). It is this handling of the sign bit outputs of LUTs  80  that corresponds to the augmented addition performed by augmented adder tree  82 . Sign correction functions  90  thus present each of the updated estimate values R i+1   mj  as updated for this, the i+1 iteration. 
     Alternatively, parity check update circuitry  56  may instead follow the approach described above in connection with Equations (14) and (15). According to this approach, LUTs  80 ,  88  for applying the W function are not used, but instead a special addition function [+] of Equation (15) is applied to the L(q mj ) values, and these values are summed according to Equation (14), to derive the updated estimate values R +1   mj  for the i+1 iteration. Variations of these parity check update approaches, and other alternative parity check update approaches, may also be realized within parity check update circuitry  56 , within LDPC decoding circuitry  38  according to this invention. The selection of the particular arithmetic approach will depend upon the available circuitry and performance of the specific implementation. 
     Referring back to  FIG. 5 , these updated estimate values R +1   mj  for the i+1 iteration are applied to memory  52 , to overwrite the previous estimate values R mj  from the prior iteration. This effectively completes the first step of the belief propagation algorithm, for this row of parity check matrix H pc , with the updating of the estimates of the column value in a row, based on the other column values involved in the same row (i.e., the same parity check equation). 
     The second step of the belief propagation algorithm, in which the input node estimates (in the form of the LLRs) are updated, begins with the application of these updated estimate values R i+1   mj  to the appropriate bit update circuit  60 , via router and reverse router circuitry  58 . The construction of router and reverse router circuitry  58  preferably depends upon the particular code arrangement, either by way of hard wiring or alternatively by way of a software controlled logic arrangement. 
       FIG. 7  illustrates an example of routing circuitry  58   f . Reverse routing circuitry  58   r  can be constructed in a similar fashion, but reversed to route signals in the opposite direction, as will be apparent to those skilled in the art having reference to this embodiment of the invention. As illustrated in  FIG. 7 , the output word of the estimates R mj  from parity check update circuitry  56  includes several values, each for one of the columns j that are involved in the current row of parity check matrix H pc . Referring back to  FIGS. 4   a  and  4   b , it is apparent that the number of columns involved in a particular row (i.e., the degree of the row) can vary. As such, the number of positions in the output word from parity check update circuitry  56  can also vary from row to row. As such, there may be instances in which one or more of the positions of the output word from parity check update circuitry  56  may be empty. 
     Routing circuitry  58   f  thus consists of a set of multiplexers  92 , which effect the forwarding of the values R mj  of the output word to the appropriate bit update circuit  60 . Knowledge of the particular code arrangement within macro matrix H defines the control of these multiplexers  92  because, according to the preferred embodiments of this invention, macro matrix H is constructed with column block grouping, by way of which only one possible column of parity check matrix H pc  is involved within each column block group, for any given row. The example of routing circuitry  58   f  illustrated in  FIG. 7  corresponds to the code shown in  FIG. 4   a , which has nine groups of column blocks, and thus involves nine positions that are applied to the nine bit update circuits  60   1  through  60   9 . 
     In this example, the left-most column block group of macro matrix H of  FIG. 4   a  has a filled “1” for every row, considering that each “1” in the matrix of  FIG. 4   a  corresponds to a sliding identity permutation matrix. As such, the left-most position of the output word from parity check update circuit  56  is always forwarded to bit update circuit  60   1 . The second-most position of the output word may be forwarded either to the second bit update circuit  60   2  or to the third bit update circuit  60   3 , depending on the code row. In any event, the control of multiplexers 92 is effected depending upon the contributions from the various column block groups to the parity check code, in each row; it is contemplated that multiplexers  92  will be switched to some extent as the process advances from one block row to another. According to this embodiment of the invention, the assignment of the positions of the output word to the various bit update circuits  60  can thus be greatly simplified with knowledge of the code, so that router circuitry  58   f  and reverse router circuitry  58   r  need not be overwhelmingly complex. In addition, this embodiment of the invention reduces the number of necessary bit update circuits  60  greatly, from what would otherwise be required (e.g., one bit update circuit for each of the k block columns of macro matrix H). 
     In any event, router circuitry  58   f  forwards the most recent iteration of estimates R i+1   mj  to the appropriate bit update circuits  60 . Bit update circuits  60  accumulate these estimates R 1+1   mj  with the estimates for the same input node in the same column j, from different rows, as will now be described relative to  FIG. 8 , which illustrates the construction of one of bit update circuits  60  according to the preferred embodiment of the invention. The others of bit update circuits  60  within LDPC decoder  38  are contemplated to be similarly constructed. 
     As shown in  FIG. 8 , bit update circuit  60  has a first adder  62  receiving an input from the router portion of router circuitry  58   f  via input aligner  63 , and a second adder  74  that forwards its output to reverse router circuitry  58   r  via output aligner  75 . Aligners  63 ,  75  are effectively shifters that can be used to align the incoming and outgoing data words as desired or necessary. Adder  62  has its output coupled to demultiplexer  64 , which forwards the output of adder  62  to a selected one of column sum memories  66 A,  66 B. Address generator circuit  68  controls the addressing of column sum memories  66 A,  66 B, and received data memory  70 . Received data memory  70  receives and stores channel input data, and applies this channel input data to an input of adder  74 ; the other input of adder  74  receives the output of a selected one of column sum memories  66 A,  66 B, via cross-switching multiplexer  72 . The other output of cross-switching multiplexer  72  is applied to a second input of adder  72 . Cross-switching multiplexer  72 , in combination with demultiplexer  64 , control the operation of column sum memories  66 A,  66 B to operate in a ping-pong buffer fashion relative to one another. One of column sum memories  66 A,  66 B is in an accumulation mode, by multiplexer  72  applying its output to adder  62  along with the adding its current value (via multiplexer  72 ) with the results from router circuitry  58 , storing the result by multiplexer  64  connecting the output of adder  62  to the input of that accumulating one of column sum memories  66 B,  66 A. Meanwhile, cross-switching multiplexer  72  is forwarding the output of the other one of column sum memories  66 A,  66 B to an input of adder  74 , to be summed with the contents of received data memory  70  and forwarded to reverse router circuitry  58   r.    
     In operation, with reference to Equation (16), column sum memories  66 A,  66 B of bit update circuits  60  accumulate the sum of the estimated values R mj  for its corresponding input node, which is associated with a corresponding one of the blocks of columns in the appropriate code. This accumulation is carried out by adder  62  receiving the most recent estimate R i+1   mj  at one input, and receiving the current accumulation of estimates R mj  for the same column j, but for different rows m, from one of column sum memories  66 A,  66 B, selected via cross-switching multiplexer  72 . Adder  62  combines these values, and forwards the sum back to the selected column sum memory  66 A,  66 B, which rewrites the accumulated sum for that row and column position, expressed as the sum 
     
       
         
           
             
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     for iteration i+1, following Equation (16). This value is retained in the selected one of column sum memories  66 A,  66 B, at a memory location associated with the corresponding input node, as addressed by address generator circuit  68 . 
     Address generator circuit  68  includes the appropriate logic and memory circuitry for maintaining and applying memory address values associated with the input nodes managed by bit update circuit  60 . According to this preferred embodiment of the invention, each of the permutation matrices involved in generating parity check matrix H pc  from macro matrix H are circularly shifted identity matrices, with the particular position of the identity diagonal varying with the position of the permutation matrix within macro matrix H. Because the rows within parity check matrix H pc  are accessed in sequence and because the permutation matrices are sliding identity matrices, address generator circuit  68  can address column sum memories  66 A,  66 B also in a sequence, using relative addressing techniques. Preferably, address generator circuit  68  includes a register for storing an offset address value for the first column position within a particular block row of macro matrix H, with each successive row then having an address that is simply incremented from this offset value, with wraparound, modulo m. For example, if the first row (m=0) has a contribution from input node column position e, the next row (m=1) will have a contribution from column position e+1, and so on. Column sum memories  66 A,  66 B can thus store the accumulated values in a linear fashion, which greatly simplifies the construction of address generation circuit  68 . Of course, the offset values will change when a new block row of macro matrix H is operated upon. 
     Alternatively, the permutation matrices within macro matrix H can be arranged differently than as circularly shifted identity matrices, so long as each permutation matrix only has a single column entry involved in each row. Address generation circuit  68  will, of course, have to comprehend the particular construction of these permutation matrices so that the memory locations in column sum memories  66  correspond to the proper column locations. 
     As mentioned above, the use of two column sum memories  66 A,  66 B, permits the updating of the acculated estimates 
     
       
         
           
             
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     for one set of columns associated with one row of parity check matrix H pc  while the accumulated estimates 
     
       
         
           
             
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     for the previous (i th ) iteration are forwarded back to parallel adder  54  for another set of columns associated with a different row of parity check matrix H pc , for example the next successive row of parity check matrix H pc  to be updated by parity check update circuitry  56 . The architecture of LDPC decoder  38  according to this preferred embodiment of the invention thus provides a high degree of performance efficiency, by way of the pipelining effected by the ping-pong arrangement of column sum memories  66 A,  66 B. In addition, this arrangement greatly improves the utilization of the memory and logic resources of LDPC decoder  38 , improving the efficiency with which it can be implemented. 
     This description will continue for the same row m and its associated columns that was updated in the prior discussion, and as such corresponds to some time in the future, after all updates have been completed for all rows in parity check matrix H pc  in iteration i+1. Cross-switching multiplexer  72  forwards the new value of accumulated sums 
     
       
         
           
             
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     from the selected one of column sum memories  66  to adder  74 . A second input to adder  74  receives the original received input node data value for the column, in the form of the ratio 
     
       
         
           
             
               
                 
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     the value r j  being the received data value, as stored in received data memory  70 . According to Equation (16), adder  74  therefore generates the current LLR value L i+1 (q j ), corresponding to the current Log Likelihood Ratio probability for the corresponding input nodes, using the value 
     
       
         
           
             
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     for iteration i+1. This value is forwarded to the reverse router circuitry  58   r , along with the values for the other corresponding columns that contribute to row m, to parallel adder  54 , in the proper order for updating of the R mj  estimates. Parallel adder  54  then generates updated R mj  estimates as described above. 
     Because the corresponding updated column LLRs L i+1 (q j ) are now available, it is convenient to use these values to evaluate this iteration i+1 to determine whether the decoding process has converged upon a valid result, satisfying the codeword equation Hc=0, GF(2). As such, the output word from reverse router circuitry  58   r  is also applied to parity check function  61 , which performs the function of “hard slicing” the current LLR values L(q mj ) to digital bit values 0 and 1, and then exclusive-ORing these bits to determine whether a 0 result is returned for that row m. It is contemplated that the circuitry or program instructions (e.g., executable by baseband processor  39 ) for performing this determination can be readily derived by those skilled in the art, having reference to this specification. The results of function  61  are retained over all rows in macro matrix H, so that the ultimate termination decision can be made. 
     This iterative process then continues, until parity check function  61  determines that the current LLRs of the codeword input nodes match a valid decoding, by returning a “0” value when applied to all of the rows of parity check matrix H pc , or until another termination criterion is met. According to the preferred embodiments of the invention, any validly decodable codeword c explicitly contains the payload data itself, i.e., the code is systematic. Baseband processor  39  or other circuitry in transceiver  20  then retrieves the payload data from the hard decision codeword, and the process is repeated for the next codeword or data block. 
     As noted above, this embodiment of the invention provides for excellent LDPC coding performance, with error rates approaching the Shannon limit, while still being readily and efficiently implementable in custom logic or in a sequence of program instructions. For example, several codes constructed in this manner have been investigated for their performance. For examples of rate ½ codes corresponding to  FIG. 4   a , bit error rate (BER) performance exceeding that of conventional LDPC codes of analogous size was obtained, resulting in BERs of on the order of 10 −5  for Eb/No values of on the order of 1.2 to 1.4 dB, depending on the size of the prime m. Similarly, for rate ⅓ codes corresponding to the example of  FIG. 4   b , improved BER performance relative to conventional LDPC codes has also been observed, with BERs on the order of 10 −5  attained at Eb/No values also on the order of 1.2 dB. In each case, the BER performance of these LDPC codes has been observed to approach that of conventional third generation (3G) cellular turbo codes, but with much more efficient implementation and performance. 
     According to a second preferred embodiment of the invention, this construction of the LDPC codes enables the use of parallel circuitry to greatly increase the performance of the LDPC decoding. More specifically, this parallelism is attained because the permutation matrices are constructed from a diagonal matrix (the identity matrix), so that incrementing of a row position within the permutation matrix means that the column position of the non-zero value also increments. The circularly (or “cyclically”) shifted identity matrix is an example of permutation matrices with this property. The parallelism effectively performs the single row parity check update and column bit update steps in parallel for multiple rows within the same row block. 
       FIG. 9  illustrates LDPC decoder  138  according to this alternative parallel architecture, for the example of a rate ½ LPDC code, where macro matrix H has twelve block rows and twenty-four block columns, as in the case of  FIG. 4   a . The functional features of LDPC decoder  138  follow that of LPDC decoder  38  described above, but with multiple instances of those functions as required to effect the parallel operation. 
     According to this embodiment of the invention, multiple R mj  memories  152  are provided. Each of memories  152  are dual port RAMs, and in this case have thirty-six output bits, and are thirty-two in number (RAMs  152   1  through  152   32 ). The number of RAMs  152  need not correspond to the number of rows of parity check matrix H pc  that are being operated in parallel, but rather is preferably selected for convenience of the integrated circuit layout and manufacturing technology; indeed, the number of RAMs  152  is entirely arbitrary. In this example, twenty-four rows of parity check matrix H pc  will be simultaneously processed, from thirty-two RAMs  152 . Accordingly, the bit width of the output of each of RAMs  152  need not match that of the number of bits to be processed for a given row. In this example, each estimate occupies six bits, and up to eight columns may contribute to a row of parity check matrix H pc . As such, the thirty-six output bits from RAM  152   1  and twelve output bits from RAM  152   2  are applied to first parallel adder block  154   1 , while the remaining twenty-four output bits from RAM  152   2  are applied to second parallel adder block  154   2 . 
     In this example, where twenty-four rows are processed in parallel, twenty-four instances of parallel adder blocks  154   1  through  154   24  are provided, each receiving eight possible R mj  values from the corresponding R RAMs  152 , along with a corresponding output of reverse router circuitry  158   r  corresponding to the LLR L(q j ) as described above. Parallel adder blocks  154  are constructed similarly as described above relative to  FIG. 5 , and each produces the individual LLRs L(q mj ) from the difference of these two inputs, for its corresponding row of parity check matrix H pc . These differences are presented to a corresponding one of twenty-four parity check update blocks  156   1  through  156   24 , for execution of the hard decision function, and the updating of the next iteration of the estimate R mj  for the columns in the operative row. 
     Parity check update blocks  156   1  through  156   24  are constructed similarly as described above relative to  FIGS. 5 and 6 , to perform the functions in the same manner as also described above. Each of parity check update blocks  156  operates independently from the others, because each is operating on a separate row of parity check matrix H pc . The outputs of parity check update blocks  156  are applied to router circuitry  58   f , and the updated estimates R mj  are applied back to the corresponding ones of RAMs  152  (this connection not shown in  FIG. 9 ), in the same manner as described above. 
     Router circuitry  158   f  routes the corresponding estimates R mj  to the appropriate bit update unit within its corresponding bit update block  160   1  through  160   9 . In this example, nine bit update blocks  160   1  through  160   9  are provided, for updating and maintaining the sums 
     
       
         
           
             
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     for the respective columns. According to this embodiment of the invention, all of the twenty-four rows being processed in parallel are from the same block row of macro matrix H, and as such the routing for all of the rows is identical. As such, the output value R mj  for corresponding bit update positions from each of parity check update blocks  156   1  through  156   24  are all routed to the same bit update block  1601 . For example, the first one of the outputs of each of parity check update blocks  156   1  through  156   24  may be routed by router circuitry  158   f  to first bit update block  160   1 , the second one of the outputs of each of parity check update blocks  156   1  through  156   24  may be routed to second bit update block  160   2 , and so on. The number of bit update blocks  160  (nine in this instance) is typically larger than the number of possible outputs from parity check update blocks  156 , and corresponds to the number of block column groups of macro matrix H. The number of bit update blocks  160  is preferably not much larger than the number of outputs from parity check update blocks  156 , however, so that router circuitry  158   f  can be relatively simple and efficient. This efficiency is a result of the construction of the LDPC code, according to the preferred embodiments of the invention, in which the order of terms, for example from left to right, in the outputs of any parity check update block  156  remains fixed after routing to bit update blocks  160 . 
     Each of bit update blocks  160  is constructed and operates similarly as described above relative to  FIG. 8 , with the exception that twenty-four bit updates are being performed in parallel. The output of bit update blocks  160 , corresponding to the L(q j ) values as described above, are then re-routed back to parity adder blocks  154   1  through  154   24  for those columns, by reverse router circuitry  158   r . In this example, the twenty-four output values output by bit update block  160   1  are routed, one each, to corresponding ones of the twenty-four parallel adder blocks  154 , and so on. In addition, reverse router circuitry  158   r  forwards these values to parity check function  161 , which performs the hard slicing and parity check sum evaluation processes to determine whether the parity check equations expressed by parity check matrix have.. have been satisfied, in which case the belief propagation decoding for the current input nodes can be terminated. 
     In the example where circularly shifted identity matrices are used in macro matrix H as described above, the memory address offset e of the first column contributing to the first row of a block row will vary from permutation matrix to permutation matrix. These changes in the address offset e must of course be considered in this embodiment of the invention, as it is in the single row case described above. However, the parallelism of this embodiment of the invention provides an additional complexity that must be comprehended by the address generator circuitry within each of bit update blocks  160 , as will now be described. 
     While the offsets e associated with each of the non-zero entries in macro matrix H will vary, the number of rows being processed in parallel is constant, namely twenty-four in this example. Because the offset e does not necessarily vary according to the parallelism multiple (e.g., twenty-four), the same input node column will likely appear in a different position from one non-zero permutation matrix entry to the next. This situation is illustrated in  FIG. 10 , with reference to data words  200  that have been routed to one of bit update blocks  160  by router circuitry  158   f . In  FIG. 10 , data word  200 ( k ) represents a parallel data word received from router circuitry  158   f , in the second embodiment of the invention described above relative to  FIG. 9 . In this example, data word  200 ( k ) includes twelve entries, and represents the R mj  estimates for twelve rows and twelve columns within a single block row, as described above. 
     However, considering the construction of the permutation matrices within macro matrix H, the column offset e for the column input node associated with the first row of a second successive block row (x+1 in  FIG. 10 ) will likely not be at a multiple of twelve from that of a prior block row (x). This circumstance is illustrated in  FIG. 10 , where the alignment of data word  200 ( k ) for block row x+1 does not align with the alignment for block row x. 
     The belief propagation algorithm requires the summing of the R mj  estimates for the same column nodes, though. Because the multiple parallel sums are being operated on simultaneously by bit update blocks  160 , the sums 
     
       
         
           
             
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     for a column node j that is updated in different block rows will thus not align, and thus will not be readily summable by bit update blocks  160 . Accordingly, the address generator circuit must contemplate this mispositioning of the columns within the parallel group. From the viewpoint of memory addressing, this problem amounts to a misalignment of the data words corresponding to the column-based values. Other hardware must also comprehend this non-alignment. An example of such hardware includes input aligner  63  and output aligner  75  as shown in  FIG. 8  for bit update blocks  60 , each of which can shift the input and output data as appropriate, and also includes temporary store registers used in such alignment and shifting. 
     Referring now to  FIG. 11  in combination with  FIG. 10 , the operation of bit update blocks  160  in managing the data word alignment to address this issue, according to this embodiment of the invention, will now be described. In process  210 , row index k is initialized to zero. This is followed by process  212 , in which data word  200 ( k ) (which, at this point, k=0) is received, for multiple rows within block row x+1. As shown in  FIG. 10 , this data word  200 ( 0 ) is misaligned, because its offset address e differs from that of block row x by eight positions. The alignment of the data word for block row x is shown in  FIG. 10 , by way of example. According to this embodiment of the invention, referring back to newly received data word  200 ( k ), data word portion  200 A(k) is that portion of data word  200 ( k ) that is ahead of the prior word boundary for block row x, and data word portion  200 B(k) is that portion of data word  200 ( k ) that is after the prior word boundary for block row x. Of course, all of bit update blocks  160  have received similar data words  200 ( k ), and will operate in similar fashion as described relative to  FIGS. 10 and 11 . 
     In process  214 , bit update block  160  retains data word portion  200 A( 0 ) in a local register or store; as mentioned above, this portion  200 A( 0 ) is that portion of data word  200 ( 0 ) that is ahead of the word boundary for block row x. In process  216 , bit update block  160  retains data word portion  200 B( 0 ) in a separate (or separately accessible) local register or store; also as mentioned above, this portion  200 B( 0 ) is that portion of data word  200 ( 0 ) that follows the block row x word boundary. 
     In process  218 , next data word  200 ( k+ 1) is then received, which, at this point, is data word  200 ( 1 ) because index k=0. Data word  200 ( k+ 1) is illustrated in  FIG. 10 , by way of example, illustrating its relative alignment to data word  200 ( k ). Data word portion  200 B(1), following the data word boundary from block row x, is retained in a local store or register in bit update block  160 , in process  220 . 
     According to this embodiment of the invention, in process  222 , data word portion  200 A( 1 ) is combined with data word portion  200 B( 0 ) that was retained in process  216 . This combined data word  200 B( 0 ): 200 A( 1 ) constitutes an aligned data word, and is aligned with the same data word alignment as that of block row x. Because, according to this embodiment of the invention, all block rows are similarly processed, this results in all data word alignments among all of the block rows being coaligned. As such, the same column input node will have a consistent data word position, over segments of rows within the block row of macro matrix H in which it participates. In process  224 , this combined data word is then processed, and its contributions to the various column input nodes summed, as described above relative to  FIG. 8 , for each data value within the combined data word. 
     In decision  225 , LDPC decoder  138  determines whether additional rows remain within the current block row (x+1) remain to be processed. If so (decision  225  is NO), index k is incremented in process  226 , and processes  218  et seq. are repeated for the next row. If the current block row is complete (decision  225  is YES), then only the initial leading data word portion  200 A( 0 ) and the most recent trailing data word portion  200 B(k) remain to be processed. These two data word portions are combined into data word  200 A( 0 ): 200 B(k+1) in process  228 , and is processed for update of the column input nodes, in similar manner as described above, in process  230 . 
     Similar management of the misaligned data words must also be performed in handling data words as they are being output by bit update circuits  160 , following the same concept as that described above for the incoming data words. It is contemplated that those skilled in the art having reference to this specification will be readily able to effectively reverse the process for the outputting of results from bit update circuits  160 . 
     In addition, another source of misalignment is also present within each permutation matrix. As discussed above, the permutation matrices are square matrices, of dimension p by p. Because p is prime, however, the parallelism factor will necessarily not evenly divide into the number of rows of the permutation matrix. This means that a partial data word will necessarily result at some point within the processing of a block row; this partial data word may occur at any point within the block row, or permutation matrix. Preferably, the overall architecture of LDPC decoder  38  is constructed and controlled to efficiently deal with these occasional partial words. It is contemplated that those skilled in the art having reference to this specification will be readily able to manage these partial word events. 
     Referring back to  FIG. 9 , while the routing and reverse routing circuitry  158   f ,  158   r , are somewhat more complex than in the single case, the decoding throughput is greatly increased by the parallelism provided by this second embodiment of the invention, by a factor corresponding to the number of parity check update blocks  156 . The ability to properly align the multiple data words is also provided, maintaining simplicity and fidelity of the bit update steps in the belief propagation method. It is contemplated that the resulting parallelism renders the present invention especially useful in high data rate applications. 
     According to further alternative embodiments of the invention, additional flexibility and memory space efficiency can be provided by arranging memory  52 ,  152  in multiple blocks. The efficiency of the memory implementations is somewhat complicated by the irregular nature of macro matrix H, considering that the weights, or degree, of the parity check rows varies from row to row; in other words, the number of columns in each row that participate in the parity check equation varies from block row to block row. In the example of  FIG. 4   a , the row degree varies between 7 and 8; for the example of  FIG. 4   b , the row degree varies between 5 and 6. Other codes may provide even a greater variation in row degree. This variation makes the memory implementation somewhat more complicated than for regular codes. 
     According to many variations of the preferred embodiments of the invention, however, the variations in row degree can be readily incorporated into the implementation and access of memories  52 ,  152 . Specifically, the following description will refer to alternative implementations of memory  52  as used for the single row-at-a-time operation described above relative to  FIGS. 5 through 8 . However, it is contemplated that those skilled in the art having reference to this specification will be readily able to implement similar arrangements in connection with multiple row processing, as described above relative to  FIGS. 9 through 11 . 
     According to one alternative implementation, memory  52  may be simply sized to correspond to the largest possible degree, times the number of bits used to store each R mj  value. For example, if the largest row degree to be contemplated is nine, and if six bits are used for each value, the width of memory  52  may be set to fifty-four bits. For those rows that have fewer than nine values, the remaining entries in the particular memory word in memory  52  may be filled with the next row&#39;s values. This approach may be generalized so that the data values for a first row of a block row of macro matrix H may be stored in a row in memory  52 ; the remainder of that row of memory  52  may be used to at least partially store data values for the next row of the block row of macro matrix H, and the remainder of the data values in that next row of the block row of macro matrix H are stored from the beginning of the next row in memory  52 . It is contemplated that the memory access control of the row values stored in memory  52  according to these alternatives can be effected by an arrangement of multiplexers and registers, as will be apparent to those skilled in the art having reference to this specification. Indeed, if the values for each row of macro matrix H fits, on the average, within a single row of memory  52 , it is contemplated that this improved packing may be implemented without involving extra memory cycles to be used in operation. 
     Further in the alternative, memory  52  may be sized to correspond to typical row degrees, but with the understanding that some rows of macro matrix H may have an extra term that does not fit within a single row of memory  52 . According to this additional alternative, an auxiliary memory may be provided to store the additional value or values for those rows of macro matrix H. This alternative approach may be particularly beneficial to provide code flexibility in LDPC decoder  38 , where memory  52  is sized to handle most code sizes but where the flexibility to handle some codes that involve large row degrees for some rows in macro matrix H is desired. 
     An important benefit of this invention is its flexibility in applying the LDPC decoding circuitry to a variety of code sizes. In general, two different approaches may be followed in order to handle a code in which the maximum row size (i.e., number of non-zero entries in a given row of parity check matrix H pc ) is greatly increased. In one approach, the operation of the decoding circuitry is time-sequenced to handle the double-sized rows in two cycles. In another approach, portions of the decoding circuitry are doubled and their results handled in parallel, permitting each row to be handled in a single cycle. According to the time-sequencing, or time doubling, approach, the construction and operation of an example of parity check update circuitry  256  will now be described relative to  FIG. 12 . As in the prior case, the input word is received from a parallel adder block, but for a single row that resides in two separate data words, the input word is necessarily processed in two cycles. As shown in  FIG. 12 , two input registers  270   a ,  270   b  are provided, for receiving a single input word for a row of parity check matrix H pc  and separating it into two input words. These two words are forwarded sequentially through look up tables  280 , and adder tree  282 . Alternatively, the feeding of input words may be pipelined and sequentially applied to a single instance of input registers  270 . 
     At the output of adder tree  282 , two-stage accumulator  284  accumulates the sequential partial sums through LUTs  280  and adder tree  282  into a full sum for the matrix row. The construction of two-stage accumulator  284  is illustrated in further detail in  FIG. 12   a . The output of adder tree  282  is applied to adder  291 , which receives the output of register  296  at another input. Multiplexer  293  receives the output of adder  291  at one input, and the output of register  296  at another input, and its output is applied to subtracting adders  286 . In operation, in the first of the two cycles for a given row, the first value is passed through adder  291  (register  296  being cleared after the prior operation, or its contribution to adder  291  being ignored for this first cycle), and is stored in register  296 . In the second of the two cycles for that row, the incoming second value is added to the first value (from register  296 ), and multiplexer  293  applies the full sum, for both portions of the row, from the output of adder  291  to subtracting adders  286 . This sum is also stored in register  296 , and is applied to subtracting adders  286  in the next cycle, for use in a second subtraction. In this way, adder tree  282  and two-stage accumulator  284  serve to generate the full sum, over the entire row of parity check matrix H pc , even though two passes through adder tree  282  is required. 
     Subtracting adders  286  receive sequential inputs from LUTs  280  to generate each subtracted value that is applied in the two input data words. As noted above, the full sum is generated over two successive cycles by adder tree  282  and two-stage accumulator  284 . Two sets of LUT  280  results are used for sequential subtractions, and as such LUT output registers  285  are provided to store the LUT outputs in sequence. Two output registers  292   a ,  292   b  are optionally provided to store the two portions of the output result, for forwarding in parallel to corresponding inputs of the bit update circuitry  60 ,  160  with registers  292   a ,  292   b  being logically in parallel with one another. Alternatively, the sequential outputs can be sequentially forwarded in two separate portions, in which the column input nodes are updated in separate cycles. 
     As a result, in this embodiment of the invention in which a single parity check update block  256  operates over two passes, the data path is made effectively wider, and capable of handling a wider set of values from the parallel adder block. This permits a great deal of flexibility in handling a wide range of code rates with the same LDPC circuitry, thus permitting programmability in the encoding and decoding processes. 
     Referring now to  FIG. 13 , the construction of the parity check update function according to an alternative implementation will now be described. In this embodiment of the invention, multiple parity check update functions  56 , in parallel, have their outputs selectively connected together to produce a full sum. This parallel connection permits wider matrix rows, wider than the width of a single parity check update function  56 , to be split across multiple datapaths and handled in a single cycle. 
     In the example of  FIG. 13 , parallel parity check update functions  56   a ,  56   b  are provided. The construction of the individual parity check update functions  56   a ,  56   b  is the same as described above relative to  FIG. 6 , and will not be repeated here. According to this alternative implementation, pair control circuit  220  is provided to selectably bus the outputs of augmented adder trees  82  to one another, so that the outputs of the individual adder trees  82  create a full sum from both adder trees as though the individual adder trees  82  were a single adder tree. This total sum is presented to each of the subtracting adders  86  in each of parity check update functions  56   a ,  56   b , so that the total sum used by each subtracting adder  86  includes not only the contributions to the sum from the augmented adder tree  82  in its own parity check update function  56 , but also the contributions to this sum from the augmented adder tree  82  in its paired parity check update function  56 . 
     Of course, more than two such parity check update functions  56  could be used in parallel to handle a single ultrawide row, if desired. For example four parity check update functions  56  could be grouped in this manner, thus providing a x4 parallel capability. In such an arrangement, it is contemplated that the corresponding control circuitry could selectably process four shorter rows at once (one from each parity check update unit), one ultrawide row by combining the sum from all four parity check update units, or a by-2 operation in which the parity check update units are paired, with each pair handling a wide row of up to 2× width. It is contemplated that the detailed construction and control of these and other alternative implementations will be apparent to those skilled in the art having reference to this specification. 
     As mentioned above, an important benefit of this invention is its great flexibility in applying the LDPC decoding circuitry to a variety of code sizes, either by time-sequencing or by the simultaneous use of multiple portions of the decoding circuitry. In the context of parallel construction such as shown in  FIG. 9 , the time-sequencing approach holds the parallelism factor Z constant, so that the circuitry operates on the same number of rows in parallel, but the circuitry is time-sequenced to handle double-sized rows in two cycles. In the other approach, the parallelism factor Z is reduced (e.g., cut in half), so that each longer row is handled in a single cycle. 
     For the time sequencing approach, referring back to  FIG. 9 , row memories  152  will supply Z partial rows with each of two successively paired words (which together are the Z full rows). In this example, parity check update blocks  156  will operate in a sequential fashion, as described above relative to  FIGS. 12 and 12   a , to process a full row of data over two cycles, and supply bit update blocks  160  with the correctly computed outputs, also over two cycles. 
     In this time sequencing approach, there may not be enough bit update blocks  160  to handle a full row, but it is contemplated that there will be at least half of the required number. Considering that each separate data item from parity check update blocks  156  originate from a different column grouping, and that each such item will likely be routed to a different bit update block  160 , bit update blocks  160  will also be time sequenced in this approach. More specifically, in the first of the two paired cycles, router circuitry  158   f  will route the first set of partial row outputs from parity check update blocks  156  to their correct bit update units  160 , and in the second of the two cycles, router circuitry  158   f  will route the second set of partial row outputs from parity check update blocks  156  to their correct bit update units  160 , which are likely to differ from those for the first set of partial row outputs. In turn, bit update units  160  will be operating on entries from one non-zero submatrix of macro matrix H in one cycle, and another non-zero submatrix in the next. In order for bit update units  160  to toggle in this fashion, it is contemplated that some circuitry, such as control circuitry and data registers, may be duplicated in bit update units  160 . 
     The construction and operation of the output paths of bit update unites  160  will similarly be constructed and operate, to effect time sequencing on the outputs passing through reverse router circuitry  158   r.    
     According to the other approach, in which the parallelism factor Z is reduced, consider an increase in the code size (row size) by a factor of two. The new parallelism factor Z′ then becomes Z/2. In each cycle, Z′ full rows will be processed in a cycle. According to this embodiment of the invention, neighboring parity check update blocks  156  are paired and combined as described above relative to  FIG. 13 , thus providing a larger effective parity check update block  156 . 
     In this approach, it is contemplated that the number of bit update blocks  160  are also less than that required by the wider row. As such, bit update blocks  160  are again time sequenced, similarly as described immediately above for the time-sequenced operation. In this approach, bit update blocks  160  must additionally have sufficient storage to be able to combine data across cycles to process the data with a higher parallelism factor (Z=2Z′), to match the data flow rate with the computation rate. 
     More specifically, partial data from each of two submatrices A, B are stored by bit update blocks  160  in a first cycle. In the next cycle, the remaining data for submatrix B are stored and the remaining data for submatrix A are grouped with the partial data for submatrix A from the previous cycle, so that this combined data appears as data having the full parallelism factor (Z=2Z′). In the next cycle, the two stored data items for submatrix B are combined and similarly processed by bit update blocks  160 ; in addition, because this cycle is the first cycle in the next cycle pair, the next row&#39;s first partial data for two submatrices A, B are stored. This pipelined-like process continues to match the full row data rate. 
     Again, the output side of bit update blocks  160  are similarly constructed to correspond to the computation rate. 
     According to these two approaches, therefore, the LDPC decoding circuitry is capable of handling a wide range of code widths. While the described examples handle increased widths up to double the size of one another, it is contemplated that this concept can be readily extended in a similar manner to greater multiples, by those skilled in the art having reference to this specification. The configuration and operation of the LDPC decoding circuitry to accomplish this change in code size is contemplated to be under software control, thus providing transceivers having a wide range of field flexibility. 
     Referring now to  FIGS. 14   a  through  14   g , alternative arrangements of memory  52  to provide even more flexibility in connection with this invention will now be described. According to these alternative arrangements, it is contemplated that memory  52  (and memory  152  in the parallel case) is implemented by multiple memory blocks. Addressing and input/output multiplexing circuitry is provided to logically arrange the memory blocks in an optimal fashion for the particular code. In this manner, the LDPC decoding circuitry according to this invention is contemplated to be widely applicable over a large set of codes, providing efficiency and excellent performance in each. 
     In the examples of  FIGS. 14   a  through  14   f , memory  252  is arranged in four blocks  252   a  through  252   d . For this purpose, it is contemplated that each block  252  has a width of q data values (which each may be six or seven bits in width), and a length (i.e., depth) of s rows, corresponding to rows in parity check matrix H pc . Routing circuitry is selectively enabled, for example under software control or in response to the state of a control register, to logically arrange the memory blocks in the manner shown in these Figures. 
     It is contemplated, relative to the examples illustrated in  FIGS. 14   a  through  14   f , that the physical arrangement of memory blocks  252   a  through  252   d  may not physically resemble any one of these arrangements in the physical sense. In addition, each of memory blocks  252   a  through  252   d  may include multiple physical memory arrays, as convenient for the physical layout. Rather, the arrangements of  FIGS. 14   a  through  14   f  represent the logical arrangement of these blocks, as effectively used by LDPC decoders according to this invention.  FIG. 14   g  illustrates an example of a physical circuit arrangement of memory blocks  252   a  through  252   d , in which logical routing circuitry  259  is illustrated as connected between memory blocks  252   a  through  252   d , and a memory bus MEMBUS. Logical routing circuitry  259  is constructed so as to logically arrange the inputs and outputs of memory blocks  252   a  through  252   d  in a selectable manner, examples of which are illustrated in  FIGS. 14   a  through  14   f . It is contemplated that logical routing circuitry  259  may be constructed by way of various multiplexers and demultiplexers, or alternatively by way of a shift register arrangement by way of which the inputs and outputs of one or more of memory blocks  252   a  through  252   d  can be selectably sequenced to memory bus MEMBUS. It is contemplated that those skilled in the art having reference to this specification will be readily able to construct the appropriate implementation of logical routing circuitry  259 . 
     In  FIG. 14   a , blocks  252   a ,  252   b ,  252   c ,  252   d  are arranged in a simple maximum width mode, with no multiplexing among the four blocks  252   a  through  252   d . In this example, the width of memory  252  is 4q, while the length of memory  252  is s, thus permitting the use of memory  252  for s rows of parity check matrix H pc . As evident from this description, both the input to and output from memory blocks  252   a  through  252   d  will be arranged in this fashion. 
       FIG. 14   b  illustrates a simple multiplexed arrangement. In this arrangement, blocks  252   a ,  252   b  are paired, and their outputs applied to one input of multiplexer  254 ; similarly, blocks  252   c ,  252   d  are paired, and have their outputs applied to the other input of multiplexer  254 , which is part of logical routing circuitry  259 . The inputs to memory blocks  252   a  through  252   d  are similarly arranged (but not shown in  FIG. 14   b  for the sake of clarity). The width of memory  252  in the logical arrangement of  FIG. 14   b  is 2q, while the length is 2s. 
       FIG. 14   c  illustrates a logical arrangement of memory blocks  252   a  through  252   d  that is the converse of that of  FIG. 14   a . In  FIG. 14   c , each of blocks  252   a ,  252   b ,  252   c ,  252   d  have their outputs coupled to respective inputs of multiplexer  254 . As such, only one of memory blocks  252   a  through  252   d  are accessed at a time. This results in a width, for memory  252 , of q values, and a length of 4s rows. Similarly, the inputs to memory blocks  252   a  through  252   d  are similarly arranged. Again, it is contemplated that the physical implementation of the logical arrangement of memory blocks  252   a  through  252   d  can be accomplished by way of logical routing circuitry  259  of  FIG. 14   g.    
     Also according to this embodiment of the invention, the data widths of blocks  252   a  through  252   d  need not be uniform among one another.  FIG. 14   d  illustrates a simple maximum width arrangement of blocks  252   a ,  252   b  (each with a width of seven) and blocks  252   c ,  252   d , with widths of three and four, respectively. In this arrangement, similarly as  FIG. 14   a , a maximum width (q=21) is provided, with minimum length (s rows). 
       FIG. 14   e  illustrates another arrangement of memory  252 , in which many of the rows are of one of two degrees (e.g., ten and eleven). Block  252   a  and block  252   c  are paired with one another, and their combined outputs of width ten are applied to an input of multiplexers  254 . Blocks  252   b ,  252   d  are similarly paired, and their combined outputs (width of eleven) are applied to a second input of multiplexers  254 . In this manner, multiplexers  256  applies either rows of ten or eleven elements to the downstream functionality. In addition, auxiliary memory  256  is also provided, for storing data elements for a relatively small number of rows that have a degree higher than eleven. The output of auxiliary memory  256  is combined with that of multiplexer  256 , to provide additional width in the row degrees, thus providing additional flexibility in the implementation. 
       FIG. 14   f  illustrates an arrangement by way of which blocks  252   a ,  252   b  both have their outputs applied to inputs of multiplexer  254 ; blocks  252   c ,  252   d  are paired, and their combined outputs are applied to a third input of multiplexer  254 . This resulting arrangement provides a data width of seven over all groups, and a common depth of s rows. 
     According to this set of alternative implementations, it is contemplated that a high degree of code flexibility is provided by this invention, in a manner that maximizes the efficiency of the memory utilization. 
     LDPC Encoding 
     As mentioned above relative to  FIG. 1 , LDPC encoder function  11  in the transmitting transceiver  10  is responsible for encoding the input datastream with the LDPC code corresponding to macro matrix H and parity check matrix H pc , as decoded by LDPC decoder circuitry  38  in receiving transceiver  20  according to the various preferred embodiments of the invention. Once the particular LDPC code is designed and arranged, having the benefits of efficient implementation in the decoder as described above, it is contemplated that those skilled in the art having reference to this specification will be readily able to implement the corresponding encoding into the transmitting tranceiver  10 , by way of conventional approaches. However, for purposes of clarity and completeness of this specification, the preferred encoding approaches will now be described. 
     As known in the art, LDPC codes do not immediately result in simple and efficient encoding methods, particularly for real world computation and hardware implementation. Structured approaches to encoder construction may or may not be available, considering that the parity check matrices according to this invention are intended to provide desirable performance characteristics on the decoding side. In general, therefore, one can consider the encoding process to be based on matrix multiplication of the an input vector x (having K information bits) with a generator matrix G to produce the codeword c: 
       c=G T x   (19)
 
     As such, a straightforward encoding process derives generator matrix G for a given macro matrix H and corresponding parity check matrix H pc , and performs the above matrix multiplication. In general, though, this requires NK multiplications (N being the number of bit nodes of codeword c), and nearly that many additions, for the encoding of a single codeword. Efficiencies in that regard are known, and will be mentioned below. 
     A preferred approach to the derivation of generator matrix G from a known parity check matrix will now be described. It is known in the art that: 
         H   pc   ·G   T =0   (20)
 
     By placing matrices G, H pc  in systematic form, the derivation of one from the other is simple. For systematic matrices G T   sys  and H sys : 
     
       
         
           
             
               
                 
                   
                     
                       H 
                       sys 
                     
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                       G 
                       sys 
                       T 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           P 
                            
                           
                             I 
                             M 
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             I 
                             K 
                           
                           P 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           PI 
                           K 
                         
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                             I 
                             M 
                           
                            
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                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     where the I z  matrices are identity matrices of dimension z. Conversion of parity check matrix H pc  into its systematic form is quite straightforward, especially using the MATLAB computing environment available from The Mathworks, Inc., by adding (modulo 2) a copy of any row to another row, and by swapping any two columns; neither operation changes the code or rank of the matrix. A simple modification of the row reduction function of MATLAB can also be used to form an identity matrix at the front of parity check matrix, by using row additions until the process is finished or a zero pivot is reached. If a zero pivot is found, column swaps are performed on the remaining matrix to locate a non-zero pivot. These operations result in the derivation of matrix P. 
     Once matrix P is derived, then generator matrix G can be readily produced using equation (21). The partitioning of systematic generator matrix G sys  that is apparent in equation (21) also defines a partitioning in codeword c that is generated from c=G sys   T x , i.e., the first K bits are the K information bits. Also, because of the column swapping that is done in the reducing of parity check matrix H pc , the order of the codeword bits is not correct for the original parity check matrix H pc . Because LDPC decoding circuitry  38  is based on this original form of parity check matrix H pc , the order of bits in codeword c must be reordered according to the order in systematic matrix H sys . Alternatively, this could be accomplished by reordering the rows of systematic generator matrix G sys . 
     In some cases, the result of the row reduction will cause parity check matrix H pc  to not be of full row rank, as including some number of dependent rows. The systematic matrix H sys  will simply have all zeros in its bottom rows after the reduction process. This will result in new code size parameters with a larger effective K and a smaller effective M (their sum still equaling N). Generator matrix G sys  will still be a valid generator matrix, although it will be encoding fewer information bits (and the true code rate will be slightly increased); parity check matrix H pc  will still be a valid parity check matrix, and used for decoding. 
     In practice, the derivation of generator matrix G sys  need only be done once per code, and as such this generation does not enter into the real-time encoding flow. However, as mentioned above, the actual encoding of payload data can be quite complex, if performed by way of a straightforward matrix multiplication of the datastream by generator matrix G sys . This is due in part to systematic generator matrix G sys  being typically quite dense, even though parity check matrix H pc  is quite sparse. Therefore, while this approach will properly generate the desired codewords, it is contemplated, in connection with this invention, that some amount of efficiency can be implemented in this encoding. 
     One approach follows that described in T. J. Richardson and R. L. Urbanke, “Efficient Encoding of Low-Density Parity-Check Codes”,  IEEE Trans. on Information Theory , Vol. 47, No. 2 (February 2001), pp. 638-656, incorporated herein by this reference. In this approach, in its construction, and by way of row and column reorderings, parity check matrix H pc  is arranged to be lower triangular over the upper M t ×M portion that will multiply against the parity bits in the codeword. Because M t &lt;M, this lower triangular property will end prior to the last few check equations or rows. The parity bits can then be recursively determined over the lower triangular portion, beginning with the first parity bit that is the only remaining unknown in the first parity equation in the first row of parity check matrix H pc . Beginning with this first parity bit, the second parity bit in the second row can be similarly determined, and so on. Upon reaching the end of the lower triangular portion, while more parity bits remain to be derived, this lower portion of the parity check matrix is quite small, and can be inverted to solve the remaining parity bits. Of course, this technique requires the situation in which parity check matrix H pc  can be arranged in this form. 
     Another approach is described in MacKay, et al., “Comparison of 
     Constructions of Irregular Gallager Codes”,  IEEE Trans. on Comm ., Vol. 47, No. 10 (IEEE, 1999), pp. 1449-1454, incorporated herein by this reference. This approach extends the Richardson et al. concept described immediately above, by forming a different form of lower triangular parity check matrix H pc . In this case, the parity bit portion of parity check matrix H pc  is lower triangular only over the upper right M t ×M portion. While the recursion approach cannot be followed in this form, parity check matrix H pc  can be viewed in terms of its major block matrices along natural divisions, from which sparse computations of matrix equations can be used to derive the encoding. 
     It is contemplated that those skilled in the art having reference to this specification will comprehend the general nature of the encoding process, especially considering these examples. It is also contemplated that those skilled artisans can implement the encoding functions either in terms of a general software implementation or, considering that the modulo 2 multiplications and additions can be trivially implemented in hardware by AND and XOR gates, in terms of custom logic hardware. 
     In Conclusion 
     According to all of the preferred embodiments of the invention, a hardware efficient LDPC coding and decoding method and circuitry is provided. The code involves a high degree of irregularity, thus providing excellent performance, nearing the Shannon limit in some cases. This performance is attained while enabling relatively simple and robust hardware implementation, despite the irregular codes. 
     While the present invention has been described according to its preferred embodiments, it is of course contemplated that modifications of, and alternatives to, these embodiments, such modifications and alternatives obtaining the advantages and benefits of this invention, will be apparent to those of ordinary skill in the art having reference to this specification and its drawings. It is contemplated that such modifications and alternatives are within the scope of this invention as subsequently claimed herein.