Abstract:
A system architecture for implementing a 10-bit Reed-Solomon code for detecting and correcting data errors in a single code word to protect a data block containing up to 1023 10-bit data symbols, i.e., the equivalent of up to 1278 8-bit symbols, including error check redundancy, maximizes the use of all allocated error correction overhead for an entire block of data, regardless of the particular error pattern characteristics encountered in a given system application. The architecture is particularly well suited for digital data processing and/or storage systems encountering non-bursty, (i.e., substantially random), error patterns, such is characteristic of data storage and retrieval systems employing semiconductor based memory stores. 5-bit extension field operations, (i.e., over a Galois field GF(2 5 )), generated by using the irreducible polynomial, P 32  (X)=X 5  +X 2  +1, over GF(2), are utilized to perform certain, requisite arithmetic functions over the Galois field GF(2 10 ) with a hardware-minimized error correction architecture.

Description:
FIELD OF THE INVENTION 
     This invention pertains to the field of error correction in digital data processing and storage systems and, more particularly, to methods and apparatus for employing a 10-bit Reed-Solomon code for the detection and correction of digital data errors within a block of data being transmitted between elements of a data processing and/or storage system. 
     BACKGROUND OF THE INVENTION 
     Error correction techniques and architectures are well known in digital data processing and communications systems, including systems having data storage subsystems such as magnetic, optical, or semiconductor based memory stores. Detection and, where possible, correction of erroneous data has been achieved by using an encoder circuit to construct some number of &#34;redundant&#34; m-bit error check symbols, which mathematically characterize the information in a selected block of data. The error check symbols are then appended to the data block and transferred or stored therewith. When the data block is received, or later retrieved from memory, the accuracy of the data can be evaluated by use of these appended error check symbols. For example, one methodology is to repeat the encoding process on the received or retrieved block of data, often using the same encoder, and then compare the newly derived error check symbols with those previously appended to the data block. If the newly derived error check symbols are identical to those appended to the data block, the information contained in the received data block is presumed to be error free; if not, &#34;error syndrome&#34; information is generated from the difference between the newly derived and originally appended error check symbols, which, depending upon the nature and amount of the error check information, can be used to locate the error(s) within the data block and determine the correct data values for substitution therefor. 
     The use of Reed-Solomon codes has become a prevalent methodology for performing error detection and correction in digital communications, processing and storage applications. In a Reed-Solomon code, sequential m-bit data symbols forming a data block are treated as being representative of coefficients of a polynomial in a variable, e.g., &#34;x&#34;. In particular, a sequence of k m-bit data &#34;message&#34; symbols {m 0 , m 1 , m 2 , . . . , m k-1  } are treated as a &#34;message polynomial,&#34; m(x), of degree k-1, where 
     
         m(x)=m.sub.0 x.sup.k-1 +m.sub.1 x.sup.k-2 + . . . +m.sub.k-2 x+m.sub.k-1, 
    
     An encoder divides the message polynomial, m(x), by a selected &#34;generator polynomial,&#34; g(x), to produce a &#34;remainder polynomial,&#34; r(x), having coefficients in the form of r m-bit error check symbols. 
     A Reed-Solomon generator polynomial, g(x), has the general form: ##EQU1## where α (&#34;alpha&#34;) is a primitive element of a Galois field, GF(q), whose q-1 powers, {α 0 , α 1 , α 2 , . . . , α q-2  }, exhaust the non-zero elements of the field and where J 0  is some arbitrary logarithm base a of the first root of the generator polynomial. Usually, in binary systems, q is a power of 2, i.e., q=2 m , for some m. For example, where r=4,--i.e., where four redundant error check symbols are to be generated--, the generator polynomial may be expressed in the factored form: 
     
         g(x)=(x-α.sup.J0)(x-α.sup.J0+1)(x-α.sup.J0+2)(x-α.sup.J0+3). 
    
     As can be seen, the nature of the generator polynomial, g(x), determines, among other things, the extent and complexity of the error correction code. In particular, the degree of the generator polynomial determines the number (r) of error check symbols contained in the remainder polynomial, r(x). However, the greater the degree of the polynomial that is selected for the generator polynomial, the more complex the associated encoder circuitry must be to perform the necessary computations. 
     Generally, the error check symbols are appended to the message symbols to form a Reed-Solomon &#34;code word&#34; of length n=k+r m-bit symbols, where the message and error check symbols are elements in the finite Galois field GF(2 m ). This is referred to as an &#34;n-symbol codeword,&#34; consisting of &#34;m-bit symbols,&#34; or as an example of an &#34;m-bit Reed-Solomon implementation.&#34; By definition, each code word, when considered as a polynomial, c(x), of degree n-1, where 
     
         c(x)=c.sub.0 x.sup.n-1 +c.sub.1 x.sup.n-2 + . . . +c.sub.n-2 x+c.sub.n-1, 
    
     is evenly divisible by the generator polynomial, g(x). In other words, c(x) mod g(x)=0. Since a correctly received code word, c&#39;(x), will also be evenly divisible by g(x), one well known procedure for detecting errors upon receipt or retrieval of a code word, c&#39;(x), is to divide it by g(x). If c&#39;(x) mod g(x) is zero, then c&#39;(x) is presumed to have been correctly received. If the remainder of c&#39;(x)/g(x) is non-zero, then one or more errors have occurred and an error correction routine is invoked by calculating r m-bit &#34;error syndromes,&#34; S o  . . . S r-1 , respectively, (i.e., &#34;symptoms of error&#34;), from the non-zero remainder of c&#39;(x)/g(x). 
     Several procedures are known for computing the error locations and proper values from the calculated error syndromes. These techniques involve error-locator-polynomial determination, root finding for determining the positions of the errors, and error value determination for determining the correct bit-pattern of the errors. In many error detection and correction architectures, the block of data being evaluated is temporarily stored in a buffer memory, while the error correction procedure is performed with the error syndromes generated from the received, or retrieved, code word. Known architectures may perform the error correction routines by using hardware circuitry under the control of a programmed state machine or arithmetic logic unit, by using a microprocessor under firmware control, or by some combination of both. For example, several known architectures perform the correction of a very small number of errors, typically no more than one or two, by hardware circuitry &#34;on-the-fly,&#34; i.e., without stopping or substantially slowing the bit-rate of the data blocks during a typical transfer, or retrieval, of multiple blocks of data. The correction of more than these one or two errors, if enabled by sufficient error syndrome information, is typically given to a system level microprocessor to carry out the error correction operations on an as-needed, or &#34;interrupt&#34; basis, i.e., where the data flow is temporarily slowed, or stopped. When this happens, the microprocessor is fed the error correction syndrome information for a data block having the more than one or two errors and calculates values for locating and correcting all of the errors under direction of error correction firmware. In the event the data is not correctable, i.e., where there are more errors detected in the block than can be corrected by the hardware and/or firmware error correction architecture, an error recovery procedure may be attempted, e.g., which may involve one or more retries to recover data from the storage media. 
     One general limitation of Reed-Solomon codes is that the maximum number of m-bit data symbols that can be corrected within an m-bit code word is limited to the integer portion, or &#34;floor,&#34; of the number of appended error check symbols (r) divided by two, i.e., INT(r/2). Another general limitation is that each m-bit code word is limited to a maximum length of 2 m  -1 symbols, sometimes referred to as its &#34;natural block length,&#34; including both the source data symbols and the appended error check symbols. Thus, implementations of Reed-Solomon error correction techniques must take into account these limitations. 
     For example, commonly owned U.S. Pat. No. 5,241,546, issued to Peterson et al., discloses a system architecture which employs a Reed-Solomon code to detect and correct data errors in a disk drive data storage system which handles data in blocks, or &#34;sectors,&#34; containing 512 8-bit symbols (or &#34;bytes&#34;), d 1!, d 2!, . . . , d 512!, respectively. Because the total number of 8-bit symbols that can be protected in a single 8-bit code word is 2 8  -1=255 symbols, each 512 byte data sector is divided into smaller blocks for purposes of performing error correction. In so doing, the Peterson et al. Patent also takes into account that data errors encountered in certain digital data communication and storage systems, such as in magnetic based disk drive systems, are often &#34;bursty&#34; in nature, i.e., the data errors tend to appear across consecutive bit positions within a block of data, as opposed to occurring randomly. The 512 data bytes are therefore serially divided into three separate &#34;interleaves,&#34; each interleave containing every third data byte,--i.e., with interleave 1 containing data bytes d 1!, d 4!, d 7!, . . . , d 511!, interleave 2 containing data bytes d 2!, d 5!, d 8!, . . . , d 512!, and interleave 3 containing data bytes d 3!, d 6!, d 9!, . . . , d 510!, respectively. Each interleave is then separately encoded to form a corresponding 8-bit Reed-Solomon code word. 
     By distributing successive data bytes into separate code words for purposes of error correction, the 3-interleave architecture offers the advantage of, for example, treating the occurrence of an &#34;error burst&#34; of three successive corrupted data bytes as, in effect, a single byte error occurring in each code word. In other words, because of the substantial likelihood that data errors, if any, will occur in consecutive data symbols, the 3-interleave architecture increases the probability that encountered symbol errors will be distributed evenly into separate code words, thereby facilitating correction of a greater number of total symbol errors per data sector, without requiring additional error correction capability for a given code word. 
     The advantage of an interleave architecture is lost, however, if the communication, data processing, and/or storage system application is not prone to bursty error patterns. For example, in a digital data system employing a semiconductor based memory in which error patterns occur in a substantially random manner, implementation of an interleave architecture may actually result in a substantial portion of the allocated error check redundancy being ineffective, since encountered errors are much less likely to be evenly distributed across the interleaves. Moreover, if the number of errors occurring in a single interleave exceed the error correction capability of that code word, the entire data sector may be lost, even if the total number of errors occurring sector-wide would otherwise have been within the allotted error correction capability had they been distributed evenly across all interleaves. 
     Thus, in non-bursty applications it would be desirable to fully utilize all allocated error check redundancy to perform error correction for an entire data block or sector, e.g., by using a single Reed-Solomon code word, regardless of the particular symbol size format of the data block. 
     SUMMARY OF THE INVENTION 
     The present invention provides a Reed-Solomon error correction methodology and architecture which is suited for digital data communication, processing and/or storage systems encountering substantially random error patterns, such as is characteristic of a data storage and retrieval system employing a semiconductor based memory store. 
     A general object of one aspect of the present invention is to provide a Reed-solomon error correction methodology and architecture ideally adapted to a mass production environment, which minimizes hardware components and the related production costs associated therewith. 
     A general object of another aspect of the present invention is to provide system architecture for implementing a 10-bit Reed-Solomon code for detecting and correcting data errors in a single code word to protect a data block containing up to 1023 10-bit data symbols, i.e., the equivalent of up to 1278 8-bit symbols, including error check redundancy. 
     A general object of yet another aspect of the present invention is to provide a Reed-Solomon error correction architecture which maximizes the use of all allocated error correction overhead for an entire block of data, regardless of the particular error pattern characteristics encountered in a given system application. 
     A general object of yet another aspect of the present invention is to provide a practical implementation of a 10-bit Reed-Solomon error correction code by employing 5-bit extension field operations over a Galois field (2 5 ) to perform certain requisite arithmetic functions with a hardware-minimized error correction architecture. 
     A more specific object of the hardware-minimized implementation aspect of the present invention is to provide a 10-bit Reed-Solomon error correction methodology and architecture which employs 5-bit extension field operations utilizing a Galois field GF(2 5 ) generated from the generator polynomial g(x)=x 5  +x 2  +1, over GF(2). 
     Yet another more specific object of the hardware-minimized implementation aspect of the present invention is to provide a 10-bit Reed-Solomon error correction methodology and architecture which generates the Galois field GF(2 10 ) using the irreducible polynomial P 1024  (y)=y 2  +y+1, over a Galois field GF(2 5 ) generated from the generator polynomial g(x)=x 5  +X 2  +1, over GF(2). 
     In accordance with these and other objects, features and aspects of the present invention, a data block of m-bit data symbols is formatted into a sequence of k 10-bit data symbols and passed through an encoder which constructs a preselected number r of 10-bit error check symbols, where k+r≦1023. The error check symbols are converted back into an m-bit symbol format and appended to the original m-bit data block for transport or storage. Upon being received, or later retrieved from storage, the m-bit data block, including the appended error check symbols, is again formatted as a sequence of 10-bit symbols and passed through a decoder/syndrome generator, which generates r 10-bit error syndromes. If any of the error syndromes are non-zero, thereby indicating the existence of one or more 10-bit symbol errors, the error syndromes are fed into an error correction module, which, using a Galois field arithmetic logic unit (&#34;GF-ALU&#34;) under the control of a programmed state machine, conducts 10-bit Reed-Solomon error correction operations (i.e., over a 10-bit Galois field GF(2 10 )), preferably to determine the location and correct values of up to two 10-bit symbol errors by hardware solution and up to four 10-bit symbol errors under firmware control. 
     According to one aspect of the present invention, the GF-ALU performs certain arithmetic functions, preferably including at least the 10-bit multiplication and inversion functions, respectively, by 5-bit extension field operations over a Galois &#34;sub field&#34; GF(2 5 ), generated by using the irreducible polynomial: 
     
         P.sub.32 (x)=x.sup.5 +x.sup.2 +1, over GF(2), 
    
     with each 5-bit nibble represented in a standard basis as a fourth degree polynomial of &#34;x&#34;; i.e., where a 5-bit nibble {d 4 ,d 3 ,d 2 ,d 1 ,d 0  } is represented as coefficients in the polynomial: 
     
         d.sub.4 x.sup.4 +d.sub.3 x.sup.3 +d.sub.2 x.sup.2 +d.sub.1 x+d.sub.0. 
    
     Addition of the 5-bit nibbles is preferably performed through a bit-wise XOR operation. Multiplication is preferably performed as standard polynomial multiplication, reduced by modulo P 32  (x). 
     According to another aspect of the present invention, the field GF(2 10 ) is generated using the irreducible polynomial: 
     
         P.sub.1024 (y)=y.sup.2 +y+1, over GF(2.sup.5), 
    
     with the 10 bit symbols represented in a standard basis as a sum of the powers of &#34;y&#34; over GF(2 5 ), i.e., with each 10-bit symbol treated as two 5-bit nibbles (e 1 , e 0 ), which represents e 1  y+e 0 , to perform the requisite arithmetic functions. Addition of the 10-bit symbols is preferably performed through a bit-wise XOR operation. Multiplication is preferably performed as standard polynomial multiplication, reduced by modulo P 1024  (y). In one preferred embodiment, the element α=(101)h (i.e., {0100000001} binary), serves as a primitive element of GF(2 10 ). 
     In this manner, 10-bit arithmetic operations are accomplished by concatenating the result of 5-bit extension field operations performed on the most significant and least significant 5-bit nibbles, respectively, of the respective 10-bit symbols. 
     These and other objects, aspects, advantages and features of the present invention will be more fully understood and appreciated by those skilled in the art upon consideration of the following detailed description of a preferred embodiment, presented in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     It is to be understood that the accompanying drawings are provided for the purpose of illustration only, and are not intended as a definition of the limits of the invention. The drawings illustrate both the design and utility of a preferred embodiment of the present invention, in which: 
     FIG. 1 is a functional block diagram of the host-to-memory (write) circuit of a semiconductor based data storage and retrieval system embodying aspects of the present invention; 
     FIG. 2 is a functional block diagram of the memory-to-host (read) circuit of the semiconductor based data storage and retrieval system shown in FIG. 1; 
     FIG. 3 is a flow chart depicting a preferred error correction process; 
     FIG. 4 is a block diagram illustrating the functional system architecture of a preferred error correction module; and 
     FIG. 5 is a block diagram illustrating the functional system architecture of a preferred Galois field arithmetic logic unit within the error correction module of FIG. 4. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring to FIG. 1, a host data processing system 10 packetizes a block of digital data for storage in an adjunct data storage and retrieval system 12. While particular blocks of data being transported by the host system 10 to the storage system 12 may vary in size and format with differing applications and operating system architectures, for purposes of describing the illustrated preferred embodiment, a commonly used data block size is selected, which contains 256 16-bit data symbols for a total of 4096 bits. A host interface circuit 14, such as, e.g., a SCSI or IDE interface device, receives the data block from the host system 10. The interface circuit 14 then transmits the data block through a 16:10 bit wide conversion circuit 16, which serially reformats the 256 16-bit data symbols into 410 10-bit data symbols, i.e., a total of 4100 bits, with the final four bits of the 410th data symbol comprising added zeros. The data block is also transmitted by the host interface circuit 14 directly into a holding buffer 18, without reformatting. 
     In alternate embodiments, the data block 12 may be packetized in varying bit formats by the host system 10 and/or host interface circuit 14. For example, the same size data block (i.e., 4096 bits) may be packetized in a &#34;byte-size&#34; format as 512 8-bit data symbols, or &#34;bytes,&#34; as well. In this later case, the bit-wide conversion circuit 16 would be designed to reformat the 512 8-bit data symbols into the 410 10-bit data symbols, i.e., at an 8:10 ratio instead of a 16:10 ratio. Alternatively, the data block may be originally packetized by the host system 10 in a 10-bit symbol format, which would eliminate the need for the bit-wide conversion circuit 16 within the data storage and handling system 12. 
     Whatever the original symbol bit-format of the data block, once converted into 10-bit format, the individual data symbols are transmitted through a 10-bit Reed-Solomon encoder circuit 20, which generates a preferred number r of 10-bit error check symbols based on a preferred generator polynomial, g(x), where r is equal to the polynomial degree of g(x). For purposes of describing the illustrated preferred embodiment, wherein correction of up to two 10-bit data errors by direct hardware solution and up to four 10-bit errors by firmware solution is preferred, an exemplary generator polynomial g(x) of degree r=8 has been selected, where (in factored form): 
     
         g(x)=(x+α.sup.0)(x+α.sup.1)(x+α.sup.2)(x+α.sup.3)(x+α.sup.4)**(x+α.sup.5)(x+α.sup.6)(x+α.sup.7). 
    
     While any number of known Reed-Solomon encoder circuits may be utilized to generate the error check symbols, e.g., such as an eight stage linear shift register with feedback, a combined encoder/syndrome generator circuit 20 of the type disclosed in an article entitled &#34;A Combined Reed-Solomon Encoder and Syndrome Generator with Small Hardware Complexity,&#34; by G. Fettweis and M. Hassner, p 1871-74, IEEE, 0-7803-0593-0/92 (1992), is preferably employed. An embedded system microprocessor 22 directs the combined encoder/syndrome generator circuit 20 to switch to &#34;encode mode,&#34; wherein the encoder circuit 20 divides the 410 10-bit data symbols of the data block by g(x) to produce a &#34;remainder&#34; comprising 8 10-bit error check symbols. 
     The 8 error check symbols are transmitted through a 10:16 bit wide conversion circuit 24, which serially reformats the 8 10-bit error check symbols into 5 16-bit symbols. The 5 16-bit symbols are then appended to the 256 16-bit data symbols of the original data block in buffer 18, i.e., which were transmitted directly from the host interface circuit 14, the 261 16-bit symbols forming a code word. The code word is released from buffer 18 by the system microprocessor 22 and is stored in a semiconductor based memory store 26, such as, e.g., a &#34;FLASH&#34; RAM. In alternate configurations and embodiments, the data block may be transmitted directly from the host interface circuit 14 into the memory store 26, i.e., without employing a holding buffer, with the reformatted (16-bit) error check symbols appended thereto directly in the memory store 26. 
     Referring to FIG. 2, the 261 16-bit code word is retrieved from the semiconductor memory store 26 and transmitted into a holding buffer 28, which may be the same buffer as buffer 18. The code word is also transmitted through a 16:10 bit-wide conversion circuit 30. Preferably, holding buffer 28 retains only the first 256 symbols of the code word, i.e., only the symbols comprising the original source data and not the 5 error check symbols, while all 261 symbols are passed through the 16:10 bit-wide conversion circuit 30. The bit-wide conversion circuit 30 reformats the 261*16-bit code word into a 418*10-bit codeword,--i.e., the 410 10-bit source data symbols, including the added four zeros in the 410th symbol, plus the 8 10-bit error check symbols. 
     The 418 10-bit symbol code word is passed through the combined encoder/syndrome generator 20, which, having been switched to &#34;syndrome&#34; mode by the system microprocessor 22, generates eight 10-bit error syndromes, S 0  . . . S 7 , respectively, therefrom. Other syndrome generation circuitry may alternatively be employed, such as conventional DFT circuits, or the like. The error syndromes are evaluated in the encoder/syndrome generator 20 and, if all syndromes are determined to be zero, (i.e., 80 consecutive zero bits), the retrieved data block is presumed to be error free and the system microprocessor 22 signals the holding buffer 28 to release the 256 16-bit data symbols to the host interface circuit 14. The host interface circuit 14 then transmits the data block to the host system 10 for use in its intended application. 
     If, however, one or more of the 8 10-bit error syndromes is non-zero, the syndromes are supplied to an error correction module 36 for determination of the location and correct values of the symbol(s) in error. As depicted in the flow chart in FIG. 3, the error correction module 36 executes the &#34;direct solution,&#34; or &#34;Peterson Gorenstein Zierler&#34; algorithm, for finding the location(s) and value(s) of up to two 10-bit symbol errors using a hardware solution. If the existence of more than two errors is detected during the error correction routine, the error syndromes are transferred to the system microprocessor 22, which, under firmware control, can determine the correct locations and values of up to four 10-bit symbol errors. 
     Referring to FIG. 4, the error correction module 36 operates under the control of a programmed state controller 42, which directs a specialized 10-bit Galois field arithmetic logic unit 44 (&#34;GF-ALU&#34;), to perform the requisite addition, multiplication, inversion, squaring and table look-up operations necessary to execute the error correction algorithm depicted in FIG. 3. Eight registers, R 0  -R 7 , 46a-h, respectively, are provided for holding interim results, with register load enables 48 (not shown in detail) provided to control which register is loaded with the result of the operation from the particular preceding instruction of the state controller 42. In the illustrated preferred embodiment, register R 0 , 46a, is a dedicated source operand for addition operations and register R 1 , 46b, is a dedicated source operand for multiplication operations, respectively. Registers R 4 , 46e, and R 5 , 46f, respectively, are used for storing the calculated error location (or &#34;locator&#34;) values, X 0  and (if double error) X 1 , respectively. Registers R 6 , 46g, and R 7 , 46h, respectively, are used for storing the calculated error values, Y 0  and (if double error) Y 1 , respectively. It may be possible to use a different register configuration in alternate embodiments, depending on the order of steps performed. 
     The following tables list the preferred state instructions given by the state controller 42 to the GF-ALU 44. Table 1 lists the state instructions for the &#34;Double Error Decoding&#34; process, Table 2 includes the state instructions for &#34;Double Error Checking&#34; process, and Table 3 includes the state instructions for &#34;Single Error Decoding and Checking&#34; process, respectively. In the state tables, &#34;st --  &#34; identifies each given state; &#34;src&#34; identifies the &#34;source&#34; register or error syndrome, respectively, which contains or comprises the input data to be used in the specified operation. &#34;Inst&#34; refers to the particular instruction, or mathematical operation to be performed by the GF-ALU 44, where &#34;load&#34; calls for loading the input value into a specified register, &#34;mult&#34; calls for multiplying the input value with the contents of register R 1 , &#34;sqre&#34; calls for squaring the input value, &#34;accm&#34; calls for adding the input value to the contents of register R 0 , and &#34;noop&#34; calls for a branch to another state if a specified zero or non-zero value is determined. The &#34;dec&#34; entry refers to the specific register location, if any, in which the resulting value of a given state operation is to be stored. The new contents, if any, of the registers following each state operation are also included, as is a brief comment to explain, if necessary, the operation that was performed. 
     
                                           TABLE 1__________________________________________________________________________Double Error Decodingst.sub.--   src inst    dec      R.sub.0        R.sub.1          R.sub.2            R.sub.3              R.sub.4                R.sub.5                  R.sub.6                    R.sub.7                      Comment__________________________________________________________________________00 S.sub.0 load    R.sub.1        S.sub.001 S.sub.2 mult    R.sub.0      t.sub.0         t.sub.0 = S.sub.0 S.sub.202 S.sub.1 sqre    R2    t.sub.1     t.sub.1 = S.sub.1.sup.203 R.sub.2 accm    R2    t.sub.2     t.sub.2 = S.sub.0 S.sub.2 + S.sub.1.sup.2 =                      det M.sub.2 !04    noop                 branch ir zero to single solution (st.sub.--                      4E)05 S.sub.3 mult    R.sub.0      t.sub.3         t.sub.3 = S.sub.0 S.sub.306 S.sub.1 load    R.sub.1        S.sub.107 S.sub.2 mult    R.sub.3          t.sub.4     t.sub.4 = S.sub.1 S.sub.208 R.sub.3 accm    R.sub.3          t.sub.5     t.sub.5 = S.sub.0 S.sub.3 + S.sub.1 S.sub.2 =                      numerator Λ.sub.1 !09    noop                 branch if zero to decoder fails (st.sub.--                      66)0a S.sub.3 mult    R.sub.0      t.sub.6         t.sub.6 = S.sub.1 S.sub.30b S.sub.2 sqre    R.sub.4   t.sub.7 t.sub.7 = S.sub.2.sup.20c R.sub.4 accm    R.sub.4   t.sub.8 t.sub.8 = S.sub.1 S.sub.3 + S.sub.2.sup.2 =                      numerator Λ.sub.2 !0d    noop                 branch if zero to decoder fails (st.sub.--                      66)0e R.sub.2 invt    R.sub.1        t.sub.9       t.sub.9 = 1/det M.sub.2 !0f R.sub.3 mult    R.sub.3 t.sub.10  t.sub.10 = Λ.sub.110 R.sub.4 mult    R.sub.4   t.sub.11                      t.sub.11 = Λ.sub.211 R.sub.3 invt    R.sub.1        t.sub.12      t.sub.12 = 1/Λ.sub.112 S1 mult    R.sub.7         t.sub.13                      t.sub.13 = S.sub.1 /Λ.sub.113 R.sub.1 sqre    R.sub.1        t.sub.14      t.sub.14 = 1/Λ.sub.1.sup.214 R.sub.14 mult    R.sub.1        t.sub.15      t.sub.15 = Λ.sub.2 /Λ.sub.1.sup.2                      115 R.sub.1 dbls    R.sub.1        t.sub.16      t.sub.16 = soln of y.sup.2 + y                      + Λ.sub.2 /Λ.sub.1.sup.2 ==                      0,ω16    noop                 branch if zero to decoder fails (st.sub.--                      66)17 R.sub.3 mult    R.sub.4   t.sub.17                      t.sub.17 = Λ.sub.1 ω = X.sub.018 R.sub.4 load    R.sub.0      t.sub.1719 R.sub.3 accm    R.sub.5     t.sub.18                      t.sub.18 = X.sub.0 + Λ.sub.1 = X.sub.11a S0 mult    R.sub.0      t.sub.19        t.sub.19 = ωS.sub.01b R.sub.7 accm    R.sub.0      t.sub.20        t.sub.20 = ωS.sub.0 + S.sub.1 /Λ.s                      ub.1 = Y.sub.11c    noop                 branch if zero to decoder fails (st.sub.--                      66)1d S.sub.0 accm    R.sub.6       t.sub.21                      t.sub.21 = S.sub.0 + Y.sub.1 = Y.sub.01e    noop                 branch if zero to decoder fails (st.sub.--                      66)1f R.sub.0 load    R.sub.7         t.sub.20                      store Y.sub.1__________________________________________________________________________ 
    
     
                                           TABLE 2__________________________________________________________________________Double Error Solution Checkingst.sub.--   src inst    dec      R.sub.0        R.sub.1          R.sub.2            R.sub.3              R.sub.4                R.sub.5                  R.sub.6                    R.sub.7                      Comment__________________________________________________________________________20 R.sub.4 sqre    R.sub.1        t.sub.22      t.sub.22 = X.sub.0.sup.221 R.sub.6 mult    R.sub.2          t.sub.23    t.sub.23 = X.sub.0.sup.2 Y.sub.022 R.sub.5 sqre    R.sub.1        t.sub.24      t.sub.24 = X.sub.1.sup.223 R.sub.7 mult    R.sub.3 t.sub.25  t.sub.25 = X.sub.1.sup.2 Y.sub.124 R.sub.3 load    R.sub.0      t.sub.2525 R.sub.2 accm    R.sub.0      t.sub.26        t.sub.26 = X.sub.0.sup.2 Y.sub.0                      + X.sub.1.sup.2 Y.sub.126 S.sub.2 accm    R.sub.0      t.sub.27        t.sub.27 = S.sub.2 + X.sub.0.sup.1 Y.sub.0 +                      X.sub.1.sup.2 Y.sub.127    noop                 branch if nonzero to decoder fails                      (st.sub.-- 6628 R.sub.4 load    R.sub.1        X.sub.029 R.sub.2 mult    R.sub.2          t.sub.28    t.sub.28 = X.sub.0.sup.3 Y.sub.02a R.sub.5 load    R.sub.1        X.sub.12b R.sub.3 mult    R.sub.3 t.sub.29  t.sub.29 = X.sub.1.sup.3 Y.sub.12c R.sub.3 load    R.sub.0      t.sub.292d R.sub.2 accm    R.sub.0      t.sub.30        t.sub.30 = X.sub.0.sup.3 Y.sub.0                      + X.sub.1.sup.3 Y.sub.12e S.sub.3 accm    R.sub.0      t.sub.31        t.sub.31 = S.sub.3 + X.sub.0.sup.3 Y.sub.0 +                      X.sub.1.sup.3 Y.sub.12f    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)30 R.sub.3 mult    R.sub.3           t.sub.32 = X.sub.1.sup.4 Y.sub.131 R.sub.4 load    R.sub.1        X.sub.032 R.sub.2 mult    R.sub.2          t.sub.33    t.sub.33 = X.sub.0.sup.4 Y.sub.033 R.sub.3 load    R.sub.0      t.sub.3234 R.sub.2 accm    R.sub.0      t.sub.34        t.sub.34 = X.sub.0.sup.4 Y.sub.0                      + X.sub.1.sup.4 Y.sub.135 S.sub.4 accm    R.sub.0      t.sub.35        t.sub.35 = S.sub.4 + X.sub.0.sup.4 Y.sub.0 +                      X.sub.1.sup.4 Y.sub.136    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)37 R.sub.2 mult    R.sub.2          t.sub.36    t.sub.36 = X.sub.0.sup.5 Y.sub.038 R.sub.5 load    R.sub.1        X.sub.139 R.sub.3 mult    R.sub.3 t.sub.37  t.sub.37 = X.sub.1.sup.5 Y.sub.13a R.sub.3 load    R.sub.0      t.sub.373b R.sub.2 accm    R.sub.0      t.sub.38        t.sub.38 = X.sub.0.sup.5 Y.sub.0                      + X.sub.1.sup.5 Y.sub.13c S.sub.5 accm    R.sub.0      t.sub.39        t.sub.39 = S.sub.5 + X.sub.0.sup.5 Y.sub.0 +                      X.sub.1.sup.5 Y.sub.13d    noop                 branch if nonzero to decoder fails(st.sub.--                      66)3e R.sub.3 mult    R.sub.3           t.sub.40 = X.sub.1.sup.6 Y.sub.13f R.sub.4 load    R.sub.1        X.sub.040 R.sub.2 mult    R.sub.2          t.sub.41    t.sub.41 = X.sub.0.sup.6 Y.sub.041 R.sub.3 load    R.sub.0      t.sub.4042 R.sub.2 accm    R.sub.0      t.sub.42        t.sub.42 = X.sub.0.sup.6 Y.sub.0                      + X.sub.1.sup.6 Y.sub.143 S.sub.6 accm    R.sub.0      t.sub.43        t.sub.43 = S.sub.6 + X.sub.0.sup.6 Y.sub.0 +                      X.sub.1.sup.6 Y.sub.144    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)45 R.sub.2 mult    R.sub.2          t.sub.44    t.sub.44 = X.sub.0.sup.7 Y.sub.046 R.sub.5 load    R.sub.1        X.sub.147 R.sub.3 mult    R.sub.3 t.sub.45  t.sub.45 = X.sub.1.sup.7 Y.sub.148 R.sub.3 load    R.sub.0      t.sub.4549 R.sub.2 accm    R.sub.0      t.sub.46        t.sub.46 = X.sub.0.sup.7 Y.sub.0                      + X.sub.1.sup.7 Y.sub.14a S.sub.7 accm    R.sub.0      t.sub.47        t.sub.47 = S.sub.6 + X.sub.0.sup.7 Y.sub.0 +                      X.sub.1.sup.7 Y.sub.14b    noop                 branch if nonzero to decoder fails(st.sub.--                      66)4c R.sub.2 stat    R.sub.2           set status for successful double decoding4d    noop                 branch to decoding complete (st.sub.--__________________________________________________________________________                      67) 
    
     
                                           TABLE 3__________________________________________________________________________Double Error Decodingst.sub.--   src inst    dec      R.sub.0        R.sub.1          R.sub.2            R.sub.3              R.sub.4                R.sub.5                  R.sub.6                    R.sub.7                      Comment__________________________________________________________________________4e S.sub.0 invt    R.sub.1        t.sub.48      t.sub.48 = 1/S.sub.04f    noop                 branch if zero to decoder fails (st.sub.--                      66)50 S.sub.1 mult    R.sub.1        t.sub.49      t.sub.49 = S.sub.1 /S.sub.0 = X.sub.051    noop                 branch if zero to decoder fails (st.sub.--                      66)52 S.sub.1 mult                 t.sub.50 = S.sub.1 X.sub.053 S.sub.2 accm    R.sub.0      t.sub.50        t.sub.51 = S.sub.1 X.sub.0 + S.sub.254    noop    R.sub.0      t.sub.51        branch if nonzero to decoder fails                      (st.sub.-- 66)55 S.sub.2 mult    R.sub.0      t.sub.52        t.sub.52 = S.sub.2 X.sub.056 R.sub.3 accm    R.sub.0      t.sub.53        t.sub.53 = S.sub.2 X.sub.0 + S.sub.357    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)58 S.sub.3 mult    R.sub.0      t.sub.54        t.sub.54 = S.sub.3 X.sub.059 S.sub.4 accm    R.sub.0      t.sub.55        t.sub.55 = S.sub.3 X.sub.0 + S.sub.45a    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)5b S.sub.4 mult    R.sub.0      t.sub.56        t.sub.56 = S.sub.4 X.sub.05c S.sub.5 accm    R.sub.0      t.sub.57        t.sub.57 = S.sub.4 X.sub.0 + S.sub.55d    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)5e S.sub.5 mult    R.sub.0      t.sub.58        t.sub.58 = S.sub.5 X.sub.05f R.sub.6 accm    R.sub.0      t.sub.59        t.sub.59 = S.sub.5 X.sub.0 + S.sub.660    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)61 S.sub.6 mult    R.sub.0      t.sub.68        t.sub.60 = S.sub.6 X.sub.062 S.sub.7 accm    R.sub.0      t.sub.68        t.sub.61 = S.sub.6 X.sub.0 + S.sub.763    noop                 branch if nonzero to decoder fails                      (st.sub.-- 66)64 R.sub.1 stat    R.sub.1           set status for successful single error                      decoding65    noop                 noop unconditional branch to state 6766 R.sub.0 stat    R.sub.0           set status to indicate decoder failure67 R.sub.1 load    R.sub.4   X.sub.0 load X.sub.0 value into locator output                      register68 S.sub.0 load    R.sub.6       Y.sub.0                      load Y.sub.0 value into error value output                      register69    noop                 decoder complete - idle state__________________________________________________________________________ 
    
     In the preferred state instructions set forth in Tables 1-3, respectively, there are a total of 106 possible instructions (or &#34;states&#34;) to be carried out during execution of the error correction algorithm. The state controller 42 will continue serial execution of instructions, until either a double or single error solution is completed, or until both fail, with the actual number of states depending upon which event occurs. If either the double or single error decoding process is successfully completed, both the location(s) of the 10-bit symbol error(s), X 0  and (if double error) X 1 , respectively, and the correct 10-bit data value(s) for substitution therefor, Y 0  and (if double error) Y 1 , respectively, are obtained from the respective registers R 4  -R 7 , 46e-h, by the system microprocessor 22. In the event both double and single solutions fail, the existence of more than two errors is presumed and the eight error syndromes, S 0  -S 7 , respectively, are transferred to the system microprocessor 22, which preferably can calculate the locations and correct values of up to four 10-bit symbol errors under firmware control. 
     Whether calculated by the GF-ALU 44, or by the system microprocessor 22, the corrected data location(s) and value(s), respectively, are substituted for the erroneous data value(s) by the system microprocessor 22, while the retrieved data block is still retained in the holding buffer 28. Because the location(s) and correct value(s), respectively, are calculated based on the 10-bit symbol format, the system microprocessor 22 translates the 10-bit location(s) and value(s) into the corresponding 16-bit location(s) and value(s), before substitution in the data block. The data block is then released from the holding buffer 28 to the host interface circuit 14 and host system 10, respectively, for use in its intended application. 
     Referring to FIG. 5, the state controller 42 serially invokes each new GF-ALU instruction 45 to the GF-ALU 44. A 10-bit symbol contained either in one of registers, R 0  . . . R 7 , 46a-h, respectively, or comprising a particular error syndrome, S 0  . . . S 7 , respectively, is used as the input value for the operation indicated in the GF-ALU instruction 45. The possible 10-bit arithmetic operations, i.e. over a Galois field GF(2 10 ), or &#34;GF(1024),&#34; include addition 52, which is preferably carried out by X&#39;OR operation, multiplication 54, inversion 56, and squaring 58, respectively, as indicated in the particular state instruction. A y 2  +y+C solution table look-up function 60 is also provided in the GF-ALU 44, for finding the quadratic solution(s) in the Galois field GF(2 10 ) for a given value of C in the double error decoding process. 
     After each instruction, the GF-ALU output 49 is loaded into one of registers, R 0  . . . R 7 , 46a-h, respectively, which is readied by a destination address 47 sent by the register load enables 48. The GF-ALU output 49 is occasionally checked for a zero or non-zero value by a zero detect circuit 62, in order to verify whether a particular single or double error solution attempt has succeeded or failed, respectively. If either a zero or non-zero value is detected, depending upon the particular state, a status bit or &#34;flag&#34; 66 is sent to the state controller 42, which will set the appropriate status, e.g. &#34;decoder fails,&#34; or &#34;successful double/single error decoding,&#34; to alert the system microprocessor 22 to obtain the requisite information from either the syndrome generator (if &#34;decoder fails&#34;) or the appropriate registers R 4  -R 7 , 46e-h, respectively. In certain state operations, the failure to locate a viable solution in the look-up table 60 will also trigger a status flag 66 to be sent. 
     In accordance with one aspect of the present invention, the operations of the GF-ALU 44 are preferably substantially hardware-minimized by employing 5-bit extension field operations over a Galois field GF(2 5 ), or GF(32), to assist in performing certain of the 10-bit arithmetic operations. In the illustrated preferred embodiment, the 5-bit extension field operation units include 5-bit multipliers 55, and inverters 57, respectively. 
     In the 5-bit operations, each 5-bit nibble or &#34;number&#34; is treated in a standard basis as a fourth degree polynomial, i.e., where each 5-bit number, {d 4 , d 3 , d 2 , d 1 , d 0  }, is represented as coefficients in the fourth degree polynomial, d 4  x 4  +d 3  x 3  +d 2  x 2  +d 1  x+d 0  x 0 . The Galois field GF(2 5 ), or GF(32), which is defined by an arbitrarily selected first element and a fifth degree generator polynomial, designated herein as &#34;p(x),&#34; is preferably generated based on the irreducible polynomial x 5  +x 2  +1. Its first element α 1  (i.e., excluding α 0 ), is selected to be equal to x, where x represents the binary field element in polynomial representation over GF(2), as is known in the art. The 31 possible non-zero field elements are represented by the successive powers of alpha, including α 0 . Accordingly, each successive element in the field may be determined by multiplying the preceding field element by α 1 , mod p(x). Put another way, the non-zero field elements are represented by the antilogs of the powers of α from 0 to 31, where each of the antilog values are calculated mod p(x), so that no antilog value can exceed 31, with α 31  mapping back to α 0 , (i.e., α 31  =α 0  =1). For example: 
     
         α.sup.1 =x.sup.1 mod p(x)=2.sup.1 =(binary) 00010; 
    
     
         α.sup.2 =α.sup.1 ·α.sup.1 mod p(x)=x.sup.1 ·x.sup.1 mod p(x)=x.sup.2 =00100; 
    
     
         α.sup.3 =α.sup.2 ·α.sup.1 mod (p(x)=01000; 
    
     
         α.sup.4 =α.sup.3 ·α.sup.1 mod (p(x)=10000; 
    
     
         α.sup.5 =α.sup.4 ·α.sup.1 =x.sup.4 ·x.sup.1 mod p(x)=x.sup.5 mod (x.sup.5 +x.sup.2 +1)=x.sup.2 +1=00101 (etc.). 
    
     After the GF(32) antilog table is calculated, generating a log table (&#34;base α&#34;) is simply a matter of reversing the antilog table by mapping each power of alpha with its corresponding antilog value. Hardware multipliers, inverters and adders for GF(32) with α 1  =x and p(x)=x 5  +x 2  +1 over GF(2), are maintained within the GF-ALU 44 to facilitate the 5-bit extension field operations. To facilitate further explanation of the 5-bit extension field operations, the antilog and log tables (decimal) for GF(2 5 ) are set forth in Tables 4 and 5: 
     
                       TABLE 4______________________________________Antilog Table/Elements of GF(2.sup.5).    i   α.sup.i______________________________________     0   1     1   2     2   4     3   8     4  16     5   5     6  10     7  20     8  13     9  26    10  17    11   7    12  14    13  28    14  29    15  31    16  27    17  19    18   3    19   6    20  12    21  24    22  21    23  15    24  30    25  25    26  23    27  11    28  22    29   9    30  18    31   1______________________________________ 
    
     
                       TABLE 5______________________________________Log Table for GF(2.sup.5)   i   LOG i!______________________________________    1   0/31    2   1    3  18    4   2    5   5    6  19    7  11    8   3    9  29   10   6   11  27   12  20   13   8   14  12   15  23   16   4   17  10   18  30   19  17   20   7   21  22   22  28   23  26   24  21   25  25   26   9   27  16   28  13   29  14   30  24   31  15______________________________________ 
    
     Preferably, the system microprocessor 22 is also provided with the appropriate Galois field logs and antilogs to facilitate calculations for the three and four error correction routines. 
     Once the results for GF(2 5 ) are determined, the 10-bit multiplication and inversion operations may be easily performed by 5-bit &#34;extension field&#34; operations. In particular, according to another aspect of the present invention, the 10 bit numbers are represented in a standard basis as a sum of powers of &#34;y&#34; over GF(2 5 ), with each 10-bit symbol treated as two 5-bit nibbles or &#34;numbers,&#34; (e 0 , e 1 ), which represent e 1  y+e 0 , to perform the requisite multiplication and inversion functions. Preferably, the multiplication of the 10-bit numbers is performed as standard polynomial multiplication, reduced by mod P 1024  (y), where P(y) is the irreducible polynomial: P 1024  (y)=y 2  +y+1, over GF(2 5 ). 
     By way of example, let m 0  =0101110100 and m 1  =1001101111. A 10-bit multiply of m 0  *m 1  is required. According to this aspect of the invention, it is carried out as follows: ##EQU2## 
     Each of the two 5-bit products, (i.e., ac, bc, ad, bd), may be determined by referring to the previously generated antilog and log (base α) tables, as follows: ##EQU3## 
     Addition of the five bit products is performed by standard XOR operation: 
     
         (ac+bc+ad)={11100}+{11110}+{00110}={00100}; 
    
     and 
     
         (bd+ac)={10010}+{11100}={01110}. 
    
     Concatenating the two results, m 0  *m 1  ={0010001110}. 
     To ensure full disclosure of the aforedescribed preferred embodiment, a corresponding verilog listing of the error correction module 36 is provided as follows: ##SPC1## 
     Thus, an error correction methodology and system architecture for implementing a 10-bit Reed-Solomon code has been disclosed. While the foregoing detailed description was directed to a semiconductor based data storage and retrieval system, it will be apparent to those skilled in the art that the described error correction methodology and architecture can be effectively practiced with any digital data storage system, e.g., such as magnetic or optical based memory systems, as well as with any system involving the handling, transfer, and/or storage of blocks of digital data between elements thereof, including digital communications systems, where the correction of data being transmitted or stored is required. While the present invention is particularly well suited for systems and architectures encountering non-bursty, substantially random data errors, it may be equally employed in those systems and architectures encountering bursty error patterns, as well. 
     Thus, it would be apparent to those skilled in the art that many more modifications are possible without departing from the inventive concepts herein. The invention, therefore, is not to be restricted except in the spirit of the appended claims.