Abstract:
Methods and apparatuses to encode and decode information. One apparatus may use an encoder with a relatively low complexity, capacity-achieving code. The code may allow information to be reliably transmitted and received across a noisy medium or channel.

Description:
CLAIM OF PRIORITY UNDER U.S.C §119  
       [0001]     This application claims priority to co-assigned U.S. Provisional Application No. 60/526,357, entitled “Low-Complexity Capacity-Achieving Code for Communication Systems,” filed on Dec. 1, 2003, which is incorporated by reference. 
     
    
     BACKGROUND  
       [0002]     1. Field  
         [0003]     The present invention relates generally to communication systems, and more specifically to encoding information and decoding information.  
         [0004]     2. Background  
         [0005]     Many modern digital communication systems use error-correcting codes (ECCs) to improve performance gains. An ECC encoder encodes a block of k message bits into a block of n code bits (where n&gt;k) to transmit. The number of k message bits transmitted per n code bits is known as the “code rate,” which is expressed as r=k/n. The code rate determines the efficiency of a coding scheme. The maximum achievable (largest) code rate that allows messages to be reliably transmitted across a particular noisy communication channel is known as the “channel capacity.” 
         [0006]     A plurality of information bits may be grouped together to form symbols or packets, which may be processed and transmitted across a channel.  
       SUMMARY  
       [0007]     A design goal of an error-correcting code (ECC) is to operate as close to capacity as possible, i.e., use the highest code rate possible that guarantees reliable communication. The complexity of encoding and decoding an ECC may increase system cost. Thus, another design goal of an ECC is to have low-complexity encoding and decoding methods.  
         [0008]     The present invention relates to methods and apparatuses for encoding information and decoding information. One apparatus may use an encoder with a relatively low complexity, capacity-achieving code. The code may allow information to be reliably transmitted and received across a noisy medium or channel.  
         [0009]     One embodiment of the codes described herein may have two major advantages over previous code designs. One advantage is the codes&#39; decoding complexity, which may be much lower than other coding methods when the code rate approaches the channel capacity. More specifically, the per-bit decoding complexity on a binary erasure channel (BEC) remains bounded as the gap to capacity vanishes. In comparison, previous code constructions that achieve capacity on a BEC have a per-bit decoding complexity that becomes infinite as the gap to capacity vanishes.  
         [0010]     Another advantage is the minimum information bit degree in the decoding graph may be greater than two, e.g., three. Therefore, these codes may be unconditionally stable, have fewer problems with low-weight code words, and do not suffer from error floor problems caused by degree two bit nodes.  
         [0011]     One aspect relates to a method of configuring an information encoder. The method comprises: for a plurality of information nodes, selecting a number of outputs for each information node, wherein a total number of outputs from the information nodes is greater than a total number of information nodes; selecting a permutation for the outputs of the information nodes to reach inputs of a plurality of parity check nodes, wherein a total number of information node outputs is equal to a total number of parity check node inputs; and selecting a number of inputs for each parity check node, wherein at least two parity check nodes have an unequal number of inputs.  
         [0012]     Another method comprises inputting a plurality of information symbols to a plurality of information nodes, one symbol per information node, where each symbol comprises at least one bit; at each information node, outputting the information symbol one or more times, wherein a total number of outputs from the information nodes is greater than a total number of information nodes; transferring the outputs of the information nodes to a plurality of parity check nodes, wherein a total number of information node outputs is equal to a total number of parity check node inputs, wherein at least two parity check nodes have an unequal number of inputs; and outputting values from the plurality of parity check nodes.  
         [0013]     Another aspect relates to an apparatus comprising an encoder configured to encode a plurality of information symbols to code symbols. The encoder is configured to: receive a plurality of information symbols at a plurality of information nodes, one symbol per information node, where each symbol comprises at least one bit; at each information node, output the information symbol one or more times, wherein a total number of outputs from the information nodes is greater than a total number of information nodes; transfer the outputs of the information nodes to a plurality of parity check nodes, wherein a total number of information node outputs is equal to a total number of parity check node inputs, wherein at least two parity check nodes have an unequal number of inputs; and output code symbols from the plurality of parity check nodes. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]      FIG. 1  illustrates a system with a transmit device and a receive device.  
         [0015]      FIG. 2A  shows an encoding process, which may be used by the system of  FIG. 1 .  
         [0016]      FIG. 2B  shows one embodiment of an error correction code (ECC) structure, which may be used by the system of  FIG. 1 .  
         [0017]      FIG. 3  illustrates another embodiment of a code structure.  
         [0018]      FIG. 4  illustrates an example of encoding a sequence of information bits to code bits.  
         [0019]      FIGS. 5A-5C  illustrate a log-likelihood ratio (LLR) decoding example.  
         [0020]      FIGS. 6A-6F  illustrate a decoding example.  
         [0021]      FIG. 7  illustrates a method of configuring an information encoder of  FIG. 1 . 
     
    
     DETAILED DESCRIPTION  
       [0022]     Information and signals may be represented using any of a variety of different technologies and techniques. For example, data, instructions, commands, information, signals, bits, symbols, and chips may be referenced by voltages, currents, electromagnetic waves, magnetic fields or particles, optical fields or particles, or any combination thereof.  
         [0023]     A digital signal may represent, for example, an image signal, a sound signal, a data signal, a video signal, or a multiplex of different signals. A digital signal, whatever its origin, may be coded and decoded. For example, U.S. Pat. No. 6,307,487 describes a coding and decoding system.  
         [0024]     The embodiments described herein may be applied to any type of transmission, by radio frequency or by cable. One field of application is digital information transmission with a certain degree of reliability on highly noise-ridden channels. For example, the embodiments may be implemented for the transmission and reception of signals by satellite or wireless communication systems. The embodiments may also be used for spatial transmission towards or between spaceships and/or space probes and, more generally, whenever the reliability of the decoding is of vital importance.  
         [0025]      FIG. 1  illustrates a system with a transmit device  100  and a receive device  120 , which communicate via a medium or channel  110 . The channel  110  may be a real-time channel, such as a path through the Internet or a broadcast link from a television transmitter to a television recipient or a telephone connection from one point to another.  
         [0026]     Alternatively, channel  110  may be a storage channel, such as a CD-ROM, disk drive, Web site, or the like. Channel  110  might even be a combination of a real-time channel and a storage channel, such as a channel formed when one person transmits an input file from a personal computer to an Internet Service Provider (ISP) over a telephone line. The input file is stored on a Web server and is subsequently transmitted to a recipient over the Internet.  
         [0027]     The transmit device  100  comprises an encoder  102 , a modulator  103  and a transmitter  104 . The receive device  120  comprises a receiver  122 , a demodulator  123 , and a decoder  124 . The transmit device  100  and the receive device  120  may comprise other elements in addition to or instead of the elements shown in  FIG. 1 .  
         [0028]     In one embodiment, the transmit device  100  may be a wireless communication device (also called a remote unit, an access terminal, a subscriber unit, etc.), such as a mobile phone, lap top computer, or personal digital assistant (PDA). The receive device  120  may be a base station in a communication system, such as a code division multiple access (CDMA) system. In another embodiment, the transmit device  100  may be a base station in a communication system, and the receive device  120  may be a wireless communication device.  
         [0029]     The encoder  102  and decoder  124  may use a novel, low complexity, error-correcting code (ECC) to encode and decode data. The code may achieve capacity of a binary erasure channel (BEC), such as the channel  110 , with modest (bounded) complexity. A BEC can erase a code bit with probability p and can then transmit the correct value with probability 1−p. A decoding method for this code may have a particularly simple form for a BEC.  
         [0030]     Desirable properties of a code may apply to other channels as well, as described in “On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit” by Sae-Young Chung, G. David Forney, Jr., Thomas J. Richardson, and Rüdiger L. Urbanke,  IEEE Commun. Letters,  5(2):58-60, February 2001. The code described herein is not limited to a BEC and may be used to improve communications on many other channels. Codes with desirable properties for a BEC may also be desirable for any packet erasure channel, such as an Internet packet loss channel.  
         [0000]     Code Structure  
         [0031]     Encoding and decoding of the code described herein may be understood via message passing on a sparse bipartite graph, which is described in “Low-density parity-check codes” by Robert G. Gallager,  Research Monograph  21, The M.I.T. Press, Cambridge, Mass., USA, 1963 and “The capacity of low-density parity check codes under message-passing decoding” by Thomas J. Richardson and Rüdiger L. Urbanke,  IEEE Trans. Inform. Theory,  47(2):599-618, February 2001.  
         [0032]     One embodiment of an error-correcting code (or sequence of codes) described herein may be a non-systematic, irregular repeat-accumulate (IRA) code.  
         [0033]      FIG. 2A  shows an encoding process, which may be used by the system of  FIG. 1 , as viewed as a serial concatenated code. Information bits are input to a repeat unit  250 , which repeats one or more of the information bits. A permutation unit  252  permutates the order (e.g., randomly selects an order) of the information bits and repeated information bits. A parity check unit  254  generates output bits based on selected information bits and repeated information bits. An accumulation unit  256  accumulates the output bits from the parity check unit  254 .  
         [0034]      FIG. 2B  shows one embodiment of an error correction code (ECC) structure  200 , which may be used by the system of  FIG. 1 . There are k information bit nodes (top shaded circles), which correspond to k information bits. Although only 6 information bit nodes are shown, there may be any k number of information bit nodes. Each information bit node has one or more edges (or lines, connections, paths) connected through a random or pseudo-random permutation to one or more parity-check nodes (squares). Pseudo-random refers to a distribution that is not completely random but resembles a random distribution. There are n parity-check nodes (squares) and n code bit nodes (bottom white circles), which correspond to n code bits.  
         [0035]     The code structure  200  comprises both deterministic elements and a random or pseudo-random element. The deterministic elements are the information bit and parity-check degrees, i.e., the number of edges (lines) attached to each circle and square. The random element (or permutation) is mapping between edges at each layer, i.e., exactly which information bit node is attached to which edge of a parity check node.  
         [0036]     Let q(i) be a number of edges (or lines) attached to an information bit node i for 1≦i≦k. Let s(j) be a number of edges emanating upwards from a parity check node j (for 1≦j≦n). Let E be the total number of edges attaching the information bit nodes to the parity-check nodes, such that:  
       E   =         ∑     i   =   1     k     ⁢           ⁢     q   ⁡     (   i   )         =       ∑     j   =   1     n     ⁢           ⁢     s   ⁡     (   j   )               
 
 In  FIG. 2B , E is equal to 19, but other embodiments may use other values of E. 
 
         [0037]     Consider the lth edge of the jth parity-check node (numbered from left to right), and let t(j, l) represent a number of the information bit node to which this lth edge attaches (for 1≦j≦n and 1≦l≦s(i)). These parameters (q(i), s(j), and t(j, l)) may define the entire structure of the code. The code bit nodes at the bottom of  FIG. 2B  may be attached to the parity-check nodes in a zigzag pattern.  
         [0000]     Encoder  
         [0038]     The encoder  102  may transform k information bits into n code bits using the following method. The information bits may be denoted by u( 1 ), . . . , u(k). The code bits may be denoted by x( 1 ), . . . , x(n). Both information bits and code bits can be taken from the binary alphabet {0,1 }. The encoder  102  may compute the code bits recursively using a formula:  
           x   ⁡     (   j   )       =       [       x   ⁡     (     j   -   1     )       +       ∑     l   =   1       s   ⁡     (   j   )         ⁢           ⁢     u   ⁡     (     t   ⁡     (     j   ,   l     )       )           ]     ⁢     mod   ⁢   2         ,       
 
 where x( 0 )=0 by convention. 
 
         [0039]     After encoding, all information bits and code bits are known. A “true value” of each edge may be defined to be the value of the bit node (either information bit node or code bit node) to which the edge attaches. This value is unique because each edge attaches to only one bit node (either information bit node or code bit node), and all edges in  FIG. 2B  connect bit nodes (information bit nodes or code bit nodes) to parity-check nodes.  
         [0040]      FIG. 3  illustrates another embodiment of a code structure  300 , where all information bit nodes have an equal number of edges, e.g., each information bit node has three edges.  
         [0041]      FIG. 4  illustrates an example of encoding a sequence of six information bits (0, 1, 1, 0, 1, 0) to nine code bits (1, 0, 1, 0, 1, 1, 1, 1, 0). The arrows in  FIG. 4  illustrate directions of binary values being passed from information bit nodes to the permutation block  402  to the parity check nodes  404  to the code bit nodes  406 . The permutation block  402  may provide random permutation or pseudo-random permutation of values from the information bit nodes  400  to the parity check nodes  404 . Once a permutation is selected, the encoder  102  and decoder  124  will use the same permutation.  
         [0042]     Each parity check node  406  receives one or more inputs from the permutation block  402  and outputs a value to a code bit node  406 . For example, the first parity check node  404 A receives a 1 and a 0 from the permutation block  402  and outputs a 1.  
         [0043]     An even parity-check node is represented by an empty square, where all edges entering the even parity check node from the bottom and/or top should sum (modulo-2) to zero.  
         [0044]     Each code bit node 406 (such as 406A) receives a value from a parity check node  404  (such as  404 A) and may output its value to another parity check node  404  (such as  404 B).  
         [0000]     Decoder  
         [0045]     A decoding method for the ECC described herein may be based on  FIG. 2B  (see also FIGS.  5 A- 5 C and  6 ), which may be called a decoding graph. The goal of decoding may be to recover all information bits from a subset of code bits. Each bit node (i.e., circle) in  FIG. 2B  may represent an “equality constraint,” where all edges entering a bit node must have the same (equal) true value.  
         [0046]     There are two types of parity constraints: even parity and odd parity. An even parity-check node is represented by an empty square, where all edges entering the even parity check node from the bottom and/or top should sum (modulo-2) to zero. An odd parity-check node is represented by a filled square, where all edges entering the odd parity check node from the bottom and/or top should sum (modulo-2) to one. By definition, every codeword in the code is a binary sequence that satisfies all of these constraints.  
         [0047]     If symbols (e.g., 00, 01, 10, 11) are used instead of bits, then there may be a plurality of parity constraint types. In this case, each type corresponds to all edges (e.g., A, B, C) entering the parity check summing to a particular value (e.g., D, i.e., A +B +C=D).  
         [0048]     A channel model used herein may specify that each code bit is observed as transmitted across a noisy channel. This observation may be used to initialize the code bits. After initialization, the message-passing decoder  124  may allow each node in FIG.  2 B to act as a separate processor, which receives messages, processes them, and then sends new messages. The decoding process may either terminate with a valid codeword or may be stopped after a fixed number of iterations.  
         [0000]     Decoding Data From a Binary Erasure Channel  
         [0049]     The BEC may have particularly simple message passing rules because each bit is either known or unknown, and a “known” bit is never in error. The decoder  124  may operate by removing edges from the graph in  FIG. 2B  and iteratively computing the original information bits. If the true value of any edge entering a bit node is known, then the equality constraint says that the true values of all edges entering that bit node are known because they must all be equal. If the true values of all but one of the edges entering a check node are known, then the even-parity constraint may be used to compute the true value of the last edge because it must equal the modulo-2 sum of all of the known edges. These two rules can be iteratively applied to the graph until the true values of all edges are known.  
         [0050]     The decoding method may be described as a graph reduction by using odd and even parity-check nodes.  
         [0051]      FIGS. 6A-6F  illustrate a decoding example with a received sequence of code bits (1, 0, ?, 0, 1, ?, 0, 0, ?), where “?” represents code bits erased by the channel  110 . The decoding method may first perform “initialization.” In  FIG. 6A , for each code bit received from the channel  110  (i.e., each bit that is not erased by the channel  110 ), if the code bit is a 1, the method toggles the color of each parity-check node attached to the code bit node (i.e., changing even-parity &lt;-&gt;odd-parity). For example, the method changes the color of the first two parity check nodes  600 A,  600 B from empty to filled because the first code bit is a 1.  
         [0052]     For both 1 and 0 received code bits, the method then deletes all edges attached to the known code bit nodes, as shown by the “Xs” in  FIG. 6B .  
         [0053]     The decoding method may then perform “iteration,” as shown in  FIG. 6C . For each parity check node with only one remaining edge (such as parity check node  600 B in  FIGS. 6B and 6C ), the value of the only connected bit node (code or information bit node) is 0 if the check node is an even-parity constraint (empty square) and 1 if the check node is an odd-parity constraint (filled square). Thus, the information bit node  602 B connected to filled parity check node  600 B is a 1. The method sets the connected bit node (e.g., information bit node  602 B) to its correct value (1) and deletes the edge, as shown in  FIG. 6D .  
         [0054]     If the value of the bit is a 1, the method toggles the color of each parity-check node attached to the bit node (information or code bit node), i.e., change even-parity &lt;-&gt;odd-parity. Thus, since information bit node  602 B is connected to parity check nodes  600 D and  600 G, the 1 from information bit node  602 B changes the color of parity check nodes  600 D and  600 G from empty to filled. The method deletes all edges attached to the known bit node  602 B, as shown in  FIG. 6E .  
         [0055]     After the edge between information bit node  602 B and parity check node  600 G is deleted, parity check node  600 G has only one edge remaining: the edge connected to code bit  604 F. Following the iteration method described above, the code bit node  604 F is set to 1, as shown in  FIG. 6E , and the edge between parity check node  600 G and code bit node  604 F is deleted, as shown in  FIG. 6F .  
         [0056]     The 1 at code bit node  604 F causes the color of parity check node  600 F to toggle from filled to empty, as shown in  FIG. 6F . Then the edge between code bit node  604 F and parity check node  600 F is deleted.  
         [0057]     The decoding method may then perform “termination.” When there are no parity-check nodes with only a single edge remaining, the method may terminate. If all edges in the decoding graph have been deleted, then decoding was successful, and all bit node values (information bits and code bits) are known. Otherwise, some bits remain unknown, as shown in  FIG. 6F , as decoding was unsuccessful.  
         [0000]     Decoding data from General Channels  
         [0058]     In contrast to binary erasure channels, decoding for general channels may require more complex messages to be passed between the nodes in the graph. These messages may represent a probability that the true value of an edge is a 0 or a 1. Representing this probability as a log-likelihood ratio (LLR) usually leads to a simpler decoder. Assume the transmitter  104  transmits a code bit X, which is equally probable to be 0 or 1, and a channel output Y is observed at the receiver  122 . In this case, the LLR may be defined as:  
         LLR   ⁡     (   X   )       =     log   ⁢         Pr   ⁡     (       Y   |   X     =   0     )         Pr   ⁡     (       Y   |   X     =   1     )         .           
 
 The channel statistics, Pr(Y|X), may be estimated from channel output by the receive device  120 . The decoding method may proceed by passing LLR messages around the decoding graph. 
 
         [0059]      FIGS. 5A-5B  illustrate a log-likelihood ratio (LLR) decoding example. The decoding method may first perform “initialization.” In  FIG. 5A , the LLR of each code bit  506  may be computed from channel observations. In  FIG. 5B , the LLR of each information bit  500  may be set to 0 (i.e., bit is equiprobably 0 or 1). The LLR of each edge may be set to the value of its adjacent bit node.  
         [0060]     The decoding method may then perform “check node iteration.” Each check node  504  has s edges, and C( 1 ), . . . , C(s) represent the LLRs of the input messages.  
         [0061]     The LLRs of the output messages, denoted by D( 1 ), . . . , D(s), may be represented as:  
         D   ⁡     (   j   )       =     f   ⁡     (       ∑       i   =   1     ,     i   ≠   j       s     ⁢           ⁢     f   ⁡     (     C   ⁡     (   i   )       )         )           
       where   ⁢               
         f   ⁡     (   x   )       =     -       log   ⁡     (     tanh   ⁡     (     x   2     )       )       .           
 
         [0062]     The decoding method may then perform “bit node iteration.” As shown in  FIG. 5A , each code bit node  506  has q edges, and A( 1 ), . . . , A(q) represent the LLRs of the input messages. The intrinsic LLR received from the channel  110  for a code bit node  506  may be represented as A( 0 ) for information bits A( 0 )=0. The LLRs of the output messages, denoted by B( 1 ), . . . , B(q), may be expressed as B(j)=B−A(j), where  
       B   =       ∑     i   =   0     s     ⁢       A   ⁡     (   i   )       .           
 
 A hard decision (decision to either one or zero) for this bit may be given by the sign (positive or negative) of B and may be output or stored. 
 
         [0063]     The decoding method may then perform “termination.” If the hard decision bits satisfy all of the code constraints, then the method found a codeword and can terminate successfully. If a codeword has not been found, and a maximum number of iterations is exceeded, then the method terminates unsuccessfully.  
         [0000]     Optimizing the Code Structure  
         [0064]     One aspect of the invention relates to choosing the values of q(i) and s(j). Let L m  be a fraction of information bit nodes with m edges, as shown in  FIG. 2B . Let R t  be a fraction of parity check nodes with t edges attached to the information bit nodes, as shown in  FIG. 2B . Mathematically, these fractions can be expressed as:  
         L   m     =                {   i        ⁢     q   ⁡     (   i   )         =   m          k         
 
 which is equal to the number of i&#39;s in the set 1≦j≦k such that q(i)=m and  
         R   t     =                {   j        ⁢     s   ⁡     (   j   )         =   t          n         
 
 which is equal to the number of j&#39;s in the set 1≦j≦n such that s(j)=t. A method is now described for choosing optimal values for R t  given a particular choice of the L m . First, degree distribution polynomials for R t  and L m  may be defined as:  
                 R   ⁡     (   x   )       =       ∑     t   ≥   1       ⁢       R   t     ⁢     x   t     ⁢           ⁢   and                       L   ⁡     (   x   )       =       ∑     m   ≥   1       ⁢       L   m     ⁢     x   m           ⁢                   .       
 
 Edge-perspective degree distribution polynomials may be expressed as:
 
ρ( x )= R ′( x )/ R ′( 1 ) and
 
λ( x )= L ′( x )/ L ′( 1 ),
 
 as shown in  FIG. 2B . R′(x) is the derivative of R(x), and L′(x) is the derivative of L(x):  
                     R   ′     ⁡     (   x   )       =       ∑     t   ≥   1       ⁢       tR   t     ⁢     x     t   -   1             ⁢                             L   ′     ⁡     (   x   )       =       ∑     m   ≥   1       ⁢     m   ⁢           ⁢     L   m     ⁢     x     m   -   1             ⁢                   .       
 
 Using these polynomials, an optimal R(x) for the BEC may be written as  
               R   opt     ⁡     (   x   )       =       1   p     ⁢         p     1   -   p       ⁢     h   ⁡     (   x   )           1   +       p     1   -   p       ⁢     h   ⁡     (   x   )                 ,     
     ⁢   where     ⁢               
         h   ⁡     (   x   )       =           ∫   0   x     ⁢       (     1   -       λ     -   1       ⁡     (     1   -   t     )         ⁢           )     ⁢     ⅆ   t             ∫   0   1     ⁢       (     1   -       λ     -   1       ⁡     (     1   -   t     )         ⁢           )     ⁢     ⅆ   t           .         
 
         [0065]     The power series expansion of R opt (x) may be found using a program such as Mathematica, which can manipulate symbolic mathematics. Appendix B lists one embodiment of a computer program for Mathematica configured to implement a code described herein.  
         [0066]     If q(i)=q for 1≦i≦k (i.e., q(i) is a set number for all information bit nodes), then the code may be called “information bit regular,” as shown in  FIG. 3 . In this case: 
        L m ={1 if m=q 
            {0 otherwise. 
 
 and λ(x) =x q−1  
 
 and therefore the formula for R opt  (x) may be simplified to:  
           R   opt     ⁡     (   x   )       =         1   -     q   ⁡     (     1   -   x     )       +       (     q   -   1     )     ⁢       (     1   -   x     )       q   /     (     q   -   1     )                 (     1   -   p     )     +     p   ⁡     (     1   -     q   ⁡     (     1   -   x     )       +       (     q   -   1     )     ⁢       (     1   -   x     )       q   /     (     q   -   1     )             )           .         
   
               
 
         [0069]     If all parity check nodes have the same number of edges, then the code may be called “check regular.” “Irregular” means each information bit node (or check node) has an unequal number of edges compared to other information bit nodes (or check node).  
         [0070]     For most choices of λ(x), the optimal parity-check degree sequence may be positive for all finite i≧2. Since one embodiment of the code will not have parity-check nodes with an infinite number of edges, this distribution may be truncated. Let S be a maximum number of edges that can be attached to a single parity-check node. There may be numerous ways to truncate R opt  (x), which may provide good performance. Let the total truncated weight for R opt  (x) be:  
       ɛ   =       ∑     j   =     S   +   1       ∞     ⁢           ⁢       R   j   opt     .           
 
         [0071]     One truncation method places all of the truncated weight on  
       R   1   opt       
 
 and may be defined by  
         R   1   opt     =     ɛ   .         
 
         [0072]     Another truncation method divides the truncated weight between  
         R   1   opt     ⁢           ⁢   and   ⁢           ⁢     R   S   opt         
 
 and may be defined by  
         R   1   opt     =       q   ⁢           ⁢     ɛ   /     (     S   +   q     )       ⁢           ⁢   and   ⁢           ⁢     R     S   +   1     opt       =     S   ⁢           ⁢     ɛ   /       (     S   +   q     )     .               
 
 The second truncation method may also be tuned based on the block length of the code. 
 
         [0073]     Another truncation method is defined by  
         R     S   +   1     opt     =   ɛ       
 
 and this requires that a small fraction of the information bits are also transmitted to get decoding started. 
 
         [0074]      FIG. 7  illustrates a method of configuring the information encoder  102  of  FIG. 1 . In block  700 , for a plurality of information nodes, the method selects a number of outputs for each information node, wherein a total number of outputs is greater than a total number of information nodes. In block  702 , the method selects a permutation for the outputs of the information nodes to reach inputs of a plurality of parity check nodes, wherein a total number of information node outputs is equal to a total number of parity check node inputs. In block  704 , the method selects a number of inputs for each parity check node, wherein at least two parity check nodes have an unequal number of inputs.  
         [0075]     Each node and each edge in  FIGS. 1-6C  can represent a vector or group of bits, i.e., a symbol. For example, a symbol may comprise 2048 bits. One of ordinary skill in the art would understand how to implement the graphs described above for encoding and decoding symbols.  
         [0076]     Appendix A describes mathematical proofs of properties for codes described herein.  
         [0077]     Those of skill would further appreciate that the various illustrative logical blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention.  
         [0078]     The various illustrative logical blocks, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed with a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.  
         [0079]     The steps of a method or algorithm described in connection with the embodiments disclosed herein may be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. A software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor such the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC and the ASIC may reside in a user terminal. In the alternative, the processor and the storage medium may reside as discrete components in a user terminal.  
         [0080]     The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.