Error correction method and apparatus

BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to a method and apparatus for correcting errors in encoded uncorrected data in a storage device, and more particularly to a method and apparatus using a unique implementation involving look-up tables for decoding syndromes of error indicative of at least two errors in said data.
2. Description of Background Art
The following references disclose basic theories and various methods and arrangements for correcting errors in storage devices: W. W. Peterson, Error-Correcting Codes, M.I.T. Press, 1961; R. C. Bose and D. K. Ray-Chaudhuri, "On a class of error-correcting binary group codes", Inf. Control 3, pp. 68-69, 1960; I. S. Reed and G. Solomon, "Polynominal codes over certain finite fields", J. Soc. Indust. Appl. Math 8, pp. 300-304, 1960; and R. T. Chien, "Cyclic decoding procedures for Bose-Chaudhuri-Hocquenghem codes", IEEE Trans. Inf. Theory, Vol. IT10, pp. 357-363, 1964.
In the copending application U.S. Ser. No. 781,449, filed Sept. 27, 1985 (now U.S. Pat. No. 4,706,250, issued Nov. 10, 1987), assigned to the assignee of the present invention, there is disclosed a two-level error correction code structure in an improved multibyte error-correcting subsystem. Data is formatted on a disk track into a number of subblocks, each within a respective block. There are also two sets of three subblock check bytes C.sub.1, C.sub.2 and C.sub.3. One set is associated with the even phase and the other with the odd phase, thus providing interleaved codewords. With this arrangement, the first level of correction (of subblock errors) is done on-the-fly at the storage device after being delayed one subblock, and the data is sent to the storage director for correction of any second level (block) errors. This on-the-fly correction is suitable only for systems operating in an asynchronous environment. The second (block) level correction is carried out using one additional check byte C.sub.0 for each phase at the end of the block.
The aforementioned copending application discloses hardware in a disk storage device that receives uncorrected data in real time and generates three first-level syndrome bytes and one second-level syndrome byte (corresponding to each phase). The first-level syndromes are decoded at the device into error pattern and error location information that is transmitted to a storage director.
This aforementioned application describes how the system functions if there are no errors in any subblock, and how errors are corrected if there is not more than one error per subblock and if there is more than one error per subblock. More specifically, syndromes S.sub.1, S.sub.2, S.sub.3 corresponding to each phase associated with a particular subblock codeword are held in a local memory. They are retained for further processing at the block level if that particular subblock's syndromes were not all zeros and no nonzero error pattern was generated by a first level decoder. The local memory also retains the identification of an uncorrected subblock as subblock identifier "f". At the end of the block, a second level syndrome, S.sub.0, from a second level syndrome generator and the first level syndromes S.sub.1, S.sub.2, S.sub.3 for subblock f from the local memory are processed by a second level decoder to correct two errors in subblock f.
There is a need for a less expensive, yet efficient, method and apparatus which uses unique look-up tables implemented in firmware or software to decode the syndromes for two-byte errors in a record of a predetermined number of bytes, in which each byte contains a desired preselected number of bits; e.g., two-byte errors in any one subblock of a two-level code format similar to that described in said copending application.
SUMMARY OF THE INVENTION
Toward this end and according to the invention, a method and apparatus is provided for correcting up to two byte errors in encoded uncorrected data in records of a predetermined length (such as a preidentified subblock of a block in the case of a multi-level error correction code format). The data is read from a storage device and corrected by decoding and processing four syndromes of error (S.sub.1, S.sub.2, S.sub.3, S.sub.0) that are generated by means disclosed in the aforementioned copending application. These syndromes are decoded in response to uncorrected errors in any one record by computing vectors P, Q and R, which are functions of the four syndromes. Binary numbers u and v are then calculated from these vectors by table look-up to enable calculation of a value of d from the sum of said binary numbers toward determining error locations. A value t, having a specific mathematical relation to the value of d, is then determined by table look-up and the error location values y and z are calculated from the offset of binary numbers u and v from the value t. Finally, the error patterns E.sub.y and E.sub.z are determined by table look-up.
As illustrated, and more specifically, errors in encoded uncorrected data in a disk storage device are corrected using a multiple level error correction code formatted into a block containing a number of subblocks. During a read operation, first level syndromes of error (S.sub.1, S.sub.2, S.sub.3) for each subblock and an additional second level syndrome (S.sub.0) common to all subblocks of the block are generated by hardware in the storage device. The first level syndromes are decoded by table look-up to provide first level error pattern and location information. The second level syndromes are decoded by computing vectors that are functions of the first and second level syndromes. The second level error locations and error patterns are determined by software or firmware using look-up tables. Any correction of errors, if needed, is done later by sending error information in a deferred mode.

The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of a preferred embodiment of the invention as illustrated in the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates a data format of a disk track that embodies a two-level code structure used in connection with the present invention.
FIG. 2 is a block diagram of the first and second level correcting portions of an ECC apparatus according to one embodiment of the invention showing flow of the data and depicting the hardware resident in the storage device and software resident in the storage director.

DESCRIPTION OF PREFERRED EMBODIMENT
FIG. 1 illustrates the data format of a disk track that embodies a two-level code structure implementing the present invention. As illustrated, data is recorded along a track 11, formatted into a plurality of fixed or variable length blocks 12. Each block 12 is divided into fixed length subblocks 14. As illustrated, each subblock 14 comprises two interleaved codewords 18, 19. Each codeward 18, 19 comprises 48 data byte positions and three subblock check bytes C.sub.1, C.sub.2, C.sub.3. Each block 12 thus comprises subblocks, each having 96 (two pairs of 48) data byte positions and three pairs of subblock check bytes C.sub.1, C.sub.2, and C.sub.3. In addition, four check bytes CR.sub.1 -CR.sub.4 for data integrity checking after ECC correction and one pair of check bytes C.sub.0 for second level error correction are appended to the end of each block 12 in a block check byte area 15. The manner in which the error correction check bytes C.sub.1 -C.sub.3 in each subblock 12 and check bytes CR.sub.1 -CR.sub.4 and C.sub.0 at the end of each block 12 are determined and produced forms no part of the present invention. The reader is referred to the aforementioned copending application, for a detailed explanation. In the following description of this invention, all process steps will be described for one of the two phases (even or odd); however, it is to be understood that the same steps and process are repeated for the other phase.
Referring now to FIG. 2, data from a data processing system (not shown) is sent via a control unit or storage director 20 to storage disk 21 of a disk storage device 100 for writing on a track which is formatted as shown in FIG. 1. In the writing and transfer of this data, three sets of check bytes C.sub.1, C.sub.2 and C.sub.3 are developed for each subblock by an ECC encoder 22. Block check byte C.sub.0 (and data integrity check bytes CR.sub.1 -CR.sub.4) are also developed by encoder 22. A subblock formatter 22A appends check bytes C.sub.1, C.sub.2 and C.sub.3 to each corresponding subblock. A block formatter 22B appends block check byte C.sub.0 (as well as data integrity check bytes, CR.sub.1 -CR.sub.4) at the end of the block. The formatted data is then recorded on storage disk 21. Computation and processing of the data integrity check bytes CR.sub.1 -CR.sub.4 forms no part of the present invention and is described in U.S. Pat. No. 4,703,485, issued Oct. 27, 1987, assigned to the assignee of the present invention.
In the readback process, the read data are checked by coding equations (1), (2), (4) and (5) in the aforementioned copending application in order to develop the syndromes of error in the conventional manner. Subblock check bytes C.sub.1, C.sub.2 and C.sub.3 are associated with syndromes S.sub.1, S.sub.2 and S.sub.3, while block-level check byte C.sub.0 is associated with the S.sub.0 syndrome byte.
The subscript numbers assigned to the syndromes, e.g., S.sub.0, S.sub.1, etc., are related to the particular T matrix employed to generate the respective check characters. Specifically, S.sub.0, which is developed from C.sub.0, corresponds to a conventional parity check byte. S.sub.3, on the other hand, is developed from C.sub.3 which is generated in accordance with logic that involves multiplying the input byte by a matrix T.sup.3. Syndromes S.sub.1 and S.sub.2, which correspond to check bytes C.sub.1 and C.sub.2, respectively, are similarly generated, using logic which involves matrices T.sup.1 and T.sup.2, respectively. Such logic for syndrome generation is well known and forms no part of the present invention.
During the readback process, uncorrected data is read from disk 21 to a first level syndrome generator 23 and a second-level syndrome generator 25 which generate first-level syndrome bytes S.sub.1, S.sub.2, S.sub.3 for each subblock and a second-level syndrome S.sub.0 common to all subblocks of the block. The syndrome bytes S.sub.1, S.sub.2, S.sub.3 are transmitted to a first level decoder 24 in storage director 20 for decoding into error pattern data. The software-implemented decoding process for decoder 24 is described below in a separate section entitled "First Level Decoding Process".
Briefly, a nonzero value for S.sub.1, S.sub.2 or S.sub.3 indicates an error. If a subblock has only one byte in error, its location x and error pattern E.sub.x, as determined by decoder 24, will be supplied to the software 26 for correction of the appropriate byte fetched from subblock buffer 27. After correction, this byte is restored into buffer 27. The error pattern E.sub.x will also be stored in local memory 28 for each of the subblocks that obtained first-level correction of one error. The second-level syndrome S.sub.0 will be modified by software at 29 to include the error pattern information E.sub.x corresponding to all of the corrected subblocks. When decoder 24 has received a nonzero value for S.sub.1, S.sub.2 or S.sub.3 and is unable to correct the subblock, it indicates the presence of more than one error in the subblock by providing a subblock identifier f to the local memory 28. The unprocessed syndromes S.sub.1, S.sub.2 and S.sub.3 for subblock f are also passed on to the local memory for later processing by the second-level decoder. Second-level decoding software 30 will combine syndrome S.sub.0 with the syndromes S.sub.1, S.sub.2, S.sub.3 from local memory 28 and convert these combined inputs by table look-up into outputs y and z indicative of the error locations and E.sub.y and E.sub.z indicative of the error patterns. These outputs y, z, E.sub.y, E.sub.z will be combined with the identifier f of the subblock in error for causing the bytes in error B.sub.y and B.sub.z to be corrected. The second-level error correction software 31 fetches the subblocks from the buffer 27 and delivers corrected data by providing correction of bytes B.sub.y and B.sub.z of subblock f using the error location information y,z and error patterns E.sub.y and E.sub.z.
And now the inventive concept will be explained more specifically. The basic two-level ECC scheme, as described in the aforementioned copending application has n subblocks in a block with N bytes in each subblock. The capability at the first level of decoding provides correction of up to one byte error in each of the subblocks. The capability, including the second level of decoding provides correction of up to two-byte errors in one subblock and one-byte error in all other subblocks in a block.
The basic error event is a "byte in error". A burst error may cause correlated errors in adjacent bytes; however, sufficient interleaving is assumed to effectively randomize these errors. With appropriate interleaving, all bytes are assumed equally likely to be in error as seen by the error correction code (ECC) scheme. Each byte contains a preselected number of bits m; the corresponding operations for the error correction code will be carried out in a finite field, GF(2.sup.m), of 2.sup.m elements. As illustrated, m is 8 and the finite field, GF(2.sup.8), has 256 elements.
Mathematical Background--Logarithms of Field Elements
Let G(x) denote a primitive polynominal of degree 8 with binary coefficients.
G(x)=g.sub.0 .sym.g.sub.1 x.sup.2 .sym. . . . .sym.g.sub.7 x.sup.7 .sym.x.sup.8
The companion matrix of the polynominal G(x) is defined as the following nonsingular matrix: ##EQU1##
The matrix T.sup.i denotes T multiplied by itself i times and all numbers reduced modulo-2. The matrices T, T.sup.2, T.sup.3, . . . , T.sup.255 are all distinct, and T.sup.255 is the identity matrix, which can also be written as T.sup.0. These 255 matrices represent (2.sup.8 -1) nonzero elements of GF(2.sup.8). Let .alpha. denote the primitive element of GF(2.sup.8). Then T.sup.i represents the nonzero element .alpha..sup.i for all i. The zero element is represented by the 8.times.8 all-zero matrix. The sum and product operations in GF(2.sup.8) are, then, defined by the modulo-2 matrix-sum and matrix-product operations, using these matrix representations of the field elements.
The elements of GF(2.sup.8) can also be represented by the 8-digit binary vectors. The square matrices in the above representation are very redundant. In fact, each matrix can be uniquely identified by just one of its columns (in a specific position), which can very well be used for representation of the corresponding field element without ambiguity. In particular, the first column of each 8.times.8 matrix in the above set is the commonly used 8-digit vector representation of the corresponding field element. This establishes a one-to-one correspondence between the set of all nonzero 8-digit vectors and the set of T.sup.i matrices representing the field elements .alpha..sup.i. Thus, each nonzero 8-digit vector S corresponds to a unique integer i (0.ltoreq.i.ltoreq.254) which can be regarded as its logarithm to the base .alpha..
Appendix A.1 is a table of logarithms which maps all field elements into powers of .alpha.. Appendix A.2 is a table of antilogarithms which maps integer powers of .alpha. into corresponding field elements. These tables were generated, using the following companion matrix T as the representation for the base element .alpha.: ##EQU2##
With the help of these tables, the product A.sub.1 .sym. A.sub.2 (of the two elements represented by 8-digit vectors A.sub.1 and A.sub.2) can be computed as follows: ##EQU3##
Each table requires a memory of 8.times.256 bits in which the word number or memory location expressed as 8-bit vector is the input vector. The stored 8-bit vector in that memory location represents the logarithm and the antilogarithm corresponding to the input vector in the two tables, respectively. Note that A=0 (the all-zero vector) cannot be processed using the log and antilog tables. Thus, it should be treated as a special case as follows: Multiplication by zero always produces zero and division by zero should not be permitted.
First-Level Decoding Process: Single Error Correction
As illustrated herein, decoding of first level errors can be accomplished using a software method involving table look-up operations.
A nonzero value for S.sub.1, S.sub.2 or S.sub.3 indicates the presence of error. Assume the subblock has only one byte (byte x) in error; then the read byte B.sub.x corresponds to the written byte B.sub.x as
B=B.sub.x .sym.E.sub.x (1)
where E.sub.x is the error pattern in the byte x
When the subblock has only one byte (byte x) in error, the syndromes are related to E.sub.x as
S.sub.1 =T.sup.x E.sub.x (2)
S.sub.2 =T.sup.2x E.sub.x (3)
S.sub.3 =T.sup.3x E.sub.x (4)
In this method, equations (2), (3) and (4) are viewed as relations among field elements in GF(2.sup.8). In particular, the matrix multiplication of the type T.sup.i B represents the product of field elements .alpha..sup.i and .beta., where .alpha..sup.i is represented by the first column of matrix T.sup.i, and .beta. is represented by the column vector B.
The product operation in GF(2.sup.8) has been discussed above, with reference to the log and antilog tables (Appendices A.1 and A.2 respectively) to the base .alpha., where .alpha. is a primitive field element. With the help of these tables, the error-position or location value x can be computed from Equations (2), (3) and (4) as follows:
x=(log.sub..alpha. S.sub.2 -log.sub..alpha. S.sub.1) modulo-255
Also
x=(log.sub..alpha. S.sub.3 -log.sub..alpha. S.sub.2) modulo-255 (5)
And the error pattern E.sub.x can be computed from Equations (2) and (5) as:
E.sub.x =log.sub..alpha..sup.-1 (log.sub..alpha. S.sub.1 -x) modulo-255 (6)
All terms in Equations (5) and (6) are 8-digit binary sequences. In modulo-255 computations, subtraction of an 8-digit binary number is equivalent to addition of its complement. For this, it is convenient to use an 8-digit binary adder with end-around carry, in which the 8-digit all-ones sequence (value 255) represents the number zero, and a high-order carry (value 256) is equivalent to the number one.
Each computation of x requires two references to the log table (Appendix A.1) and one modulo-255 subtract operation. Similarly, the computation of E.sub.x requires one reference to the antilog table (Appendix A.2). The byte B.sub.x is then corrected as B.sub.x +E.sub.x.
Note that if S.sub.1 =S.sub.2 =0 and S.sub.3 .noteq.0 then the error is in check byte C.sub.2. In Equation (5), x is computed two ways, the result of which must be the same; otherwise there are more than one bytes in error. Also, if S.sub.1 .noteq.S.sub.2 and S.sub.1 or S.sub.2 is zero, then there are more than one bytes in error. The subblock in that case has errors but remains uncorrected through the first level processing. Such subblock is identified by the subblock number f and the corresponding syndromes S.sub.1, S.sub.2, S.sub.3 are stored in local memory and passed on for later second level processing.
Second-Level Decoding Process: Two-Error Correction
Assume only one subblock contains two bytes in error denoted by y and z with error patterns E.sub.y and E.sub.z, respectively. The syndromes S.sub.0, S.sub.1, S.sub.2 and S.sub.3 are related to E.sub.y and E.sub.z as follows:
S.sub.0 =E.sub.y .sym.E.sub.z (7)
S.sub.1 =T.sup.y E.sub.y .sym.T.sup.z E.sub.z (8)
S.sub.2 =T.sup.2y E.sub.y .sym.T.sup.2z E.sub.z (9)
S.sub.3 =T.sup.3y E.sub.y .sym.T.sup.3z E.sub.z (10)
The first level processing for the corresponding subblock will have detected these errors as a multiple error. With S.sub.1, S.sub.2 and S.sub.3 available at the subblock-level, the subblock-level processing of syndromes will not miscorrect these errors as a one-symbol error E.sub.x in position x.
To decode the combined set of subblock and block-level syndromes for two-symbol errors, see Appendix B which explains, the theory behind the decoding algorithm. First, vectors P, Q and R are obtained. As illustrated, they are 8-digit constants which are functions of the syndromes S.sub.0, S.sub.1, S.sub.2, and S.sub.3, as given by:
P=(S.sub.2 .circle..times. S.sub.2).sym.(S.sub.3 .circle..times. S.sub.1) (11)
Q=(S.sub.2 .circle..times. S.sub.1).sym.(S.sub.3 .circle..times. S.sub.0) (12)
R=(S.sub.0 .circle..times. S.sub.2).sym.(S.sub.1 .circle..times. S.sub.1) (13)
where .circle..times. denotes the product operation of the field elements in GF(2.sup.8), and the field elements are represented by binary 8-digit vectors. The product operation can be realized using hard-wired logic or through the use of log and antilog tables in GF(2.sup.8).
Note that P, Q, and R are necessarily nonzero when there are two bytes in error and both are in data bytes of the subblocks. In contrast, when the check byte C.sub.1 or C.sub.2 is among the two erroneous bytes, this is indicated by P=0 and R=0, respectively.
Assume now that there are exactly two erroneous bytes in one of the subblocks. Then the error-location values y and z are two unique solutions of i in the equation:
T.sup.-2i P.sym.T.sup.-i Q=R (14)
where P, Q, and R are functions of the syndromes S.sub.0, S.sub.1, S.sub.2, and S.sub.3, as given by Equations (11) to (13).
For each of the two solution values of i, the error pattern is given by:
E.sub.i =R/(T.sup.2i S.sub.0 .sym.S.sub.2) (15)
The proof of this appears in Appendix B.
According to an important feature of the invention, decoding of the combined set of subblock and block-level syndromes for two-symbol errors can be accomplished using a software method involving table look-up operations. Vectors P, Q and R are computed from syndromes S.sub.0, S.sub.1, S.sub.2 and S.sub.3, using the log and antilog tables of Appendices A.1 and A.2, respectively. This requires, at the most, eighteen references to the tables in memory, six binary-add (modulo-255) operations, and three vector-add (modulo-2) operations.
The error-location values y and z can be obtained through a simple table look-up procedure. The table and the theory behind this procedure appear in Appendix C. In this method, the error-location values y and z are obtained through the following four-step procedure.
Step 1: Obtain two binary numbers u and v from vectors P, Q, and R, using log and antilog tables Appendices A.1 and A.2, respectively.
u=(log.sub..alpha. P-log.sub..alpha. Q) modulo-255 (16)
v=(log.sub..alpha. R-log.sub..alpha. Q) modulo-255 (17)
Step 2: Calculate the value d from the sum of said two binary numbers as a step toward determining the locations of the two errors, as follows:
d=(u+v) modulo-255 (18)
Step 3: Obtain the value t, having a specific mathematical relation to value d, from the table of Appendix C.1.
Step 4: Obtain error-location values y and z by calculating the offset of said binary numbers from the value t, as follows:
y=(u-t) modulo-255 (19)
z=(t-v) modulo-255 (20)
All terms in Equations (16) to (20) of the above procedure are 8-digit binary sequences undergoing modulo-255 add or subtract operations. The procedure requires four table look-up operations, four modulo-255 subtact operations, and one modulo-255 add operation. In this procedure, an invalid value of d (the one with no entry in Appendix C.1) or an invalid error location value for y or z (greater than m+1) indicates an uncorrectable error involving three or more bytes in error.
The error pattern E.sub.y can be computed using the log and antilog tables (Appendices A.1 and A.2, respectively) in accordance with Equation (15), in which matrix multiplication T.sup.2y S.sub.0 is replaced by the corresponding field element given by the product .alpha..sup.2y .circle..times. S.sub.0 of two field elements.
The error pattern E.sub.z can be computed similarly, using Equation (15) or alternatively from Equation (9), which gives:
E.sub.z =S.sub.0 .sym.E.sub.y (21)
The subblock error correction is then accomplished by correcting bytes B.sub.y and B.sub.z with error patterns E.sub.y and E.sub.z.
While the embodiment, as illustrated, assumed a two-level code structure in which a one-byte error in a codeword is correctable at the first level and a two-byte error in a codeword is correctable at the block level, it should be understood that the method and apparatus may be used to correct two byte errors in any single or multi-level code structure.
It should also be recognized that the disclosed method and apparatus will also operate to decode the syndromes for two-byte errors in a record of a predetermined number of bytes, in which each byte contains a preselected number of bits.
Also, while the method and apparatus, as disclosed, correct errors in encoded uncorrected data in a magnetic disk storage device, they are equally applicable to a tape or optical storage device.
Finally, it should be understood that, if preferred, in lieu of the software implementation herein described, decoding of first level (single) errors may be accomplished by hardware, such as disclosed in U.S. Pat. No. 4,525,838, issued June 25, 1985, assigned to the assignee of the present invention.
It will therefore be understood by those skilled in the art that the foregoing and other applications and/or implementations may be made to the method and apparatus herein described without departing from the spirit, scope and teaching of the invention. Accordingly, the method and apparatus herein disclosed are to be considered merely as illustrative and the invention is to be limited only as specified in the claims.
While the invention has been particularly shown and described with reference to a preferred embodiment thereof, it will be understood by those skilled in the art that various other changes in the form and details may be made therein without departing from the spirit and scope of the invention.
APPENDEX A.1__________________________________________________________________________Logarithms (base .alpha.) in GF (2.sup.8)log.sub..alpha. S = iS i S i S i S i S i__________________________________________________________________________001 00000000 052 10111100 103 11111010 154 10100101 205 01000000002 00000001 053 11111100 104 10111101 155 11110100 206 11111011003 00001101 054 11001101 105 10111000 156 10101101 207 00110000004 00000010 055 00110001 106 11111101 157 10100110 208 10111110005 00011010 056 10110101 107 00011000 158 10000100 209 10010000006 00001110 057 11010100 108 11001110 159 11110101 210 10111001007 10110010 058 10011111 109 10011100 160 00011111 211 00100110008 00000011 059 10010111 110 00110010 161 01001010 212 11111110009 10111111 060 00101001 111 01100011 162 01110110 213 00001100010 00011011 061 01111100 112 10110110 163 11100100 214 00011001011 10010001 062 00111011 113 10011010 164 00010110 215 10110001012 00001111 063 01110010 114 11010101 165 01100001 216 11001111013 10111010 064 00000110 115 01111000 166 01010010 217 11001011014 10110011 065 01111111 116 10100000 167 11111000 218 10011101015 00100111 066 01001000 117 11101000 168 01101000 219 00111001016 00000100 067 01011111 118 10011000 169 00001000 220 00110011017 00110100 068 00110110 119 11100110 170 01001111 221 11101110018 11000000 069 00100011 120 00101010 171 01011010 222 01100100019 11101111 070 01101110 121 10100010 172 11011111 223 11000110020 00011100 071 10001001 122 01111101 173 10000001 224 10110111021 01100101 072 11000010 123 00100001 174 01011101 225 00010111022 10010010 073 00101100 124 00111100 175 10000111 226 10011011023 11000111 074 01010111 125 11101010 176 10010101 227 01100010024 00010000 075 11100001 126 01110011 177 01110000 228 11010110025 11010000 076 11110001 127 01001100 178 00111111 229 01010011026 10111011 077 10100100 128 00000111 179 00101111 230 01111001027 11001100 078 10101100 129 01011001 180 11011011 231 11111001028 10110100 079 10000011 130 10000000 181 10001011 232 10100001029 10011110 080 00011110 131 10000110 182 01101011 233 00100000030 00101000 081 01110101 132 01001001 183 10101001 234 11101001031 00111010 082 00010101 133 11100011 184 11001010 235 01001011032 00000101 083 01010001 134 01100000 185 00111000 236 10011001033 01000111 084 01100111 135 11110111 186 11101101 237 01110111034 00110101 085 01001110 136 00110111 187 11000101 238 11100111035 01101101 086 11011110 137 11000100 188 10001111 239 11100101036 11000001 087 01011100 138 00100100 189 00100101 240 00101011037 01010110 088 10010100 139 10101111 190 00001011 241 11100000038 11110000 089 00111110 140 01101111 191 10110000 242 10100011039 10101011 090 11011010 141 00101110 192 00010011 243 10000010040 00011101 091 01101010 142 10001010 193 11011100 244 01111110041 00010100 092 11001001 143 10101000 194 11011000 245 01011110042 01100110 093 11101100 144 11000011 195 10001100 246 00100010043 11011101 094 10001110 145 10101110 196 01000110 247 10001000044 10010011 095 00001010 146 00101101 197 01101100 248 00111101045 11011001 096 00010010 147 10100111 198 01010101 249 01101001046 11001000 097 11010111 148 01011000 199 10101010 250 11101011047 10001101 098 01000101 149 10000101 200 11010011 251 00001001048 00010001 099 01010100 150 11100010 201 10010110 252 01110100049 01000100 100 11010010 151 11110110 202 01111011 253 01010000050 11010001 101 01111010 152 11110010 203 01110001 254 01001101051 01000001 102 01000010 153 11110011 204 01000011 255 01011011__________________________________________________________________________ Note: S in polynomial notation has high order term on the left.
APPENDIX A.2__________________________________________________________________________Antilogarithms (base .alpha.) in GF (2.sup.8).alpha..sup.i = Si S i S i S i S i S__________________________________________________________________________000 00000001 051 11011100 102 00101010 153 11101100 204 00011011001 00000010 052 00010001 103 01010100 154 01110001 205 00110110002 00000100 053 00100010 104 10101000 155 11100010 206 01101100003 00001000 054 01000100 105 11111001 156 01101101 207 11011000004 00010000 055 10001000 106 01011011 157 11011010 208 00011001005 00100000 056 10111001 107 10110110 158 00011101 209 00110010006 01000000 057 11011011 108 11000101 159 00111010 210 01100100007 10000000 058 00011111 109 00100011 160 01110100 211 11001000008 10101001 059 00111110 110 01000110 161 11101000 212 00111001009 11111011 060 01111100 111 10001100 162 01111001 213 01110010010 01011111 061 11111000 112 10110001 163 11110010 214 11100100011 10111110 062 01011001 113 11001011 164 01001101 215 01100001012 11010101 063 10110010 114 00111111 165 10011010 216 11000010013 00000011 064 11001101 115 01111110 166 10011101 217 00101101014 00000110 065 00110011 116 11111100 167 10010011 218 01011010015 00001100 066 01100110 117 01010001 168 10001111 219 10110100016 00011000 067 11001100 118 10100010 169 10110111 220 11000001017 00110000 068 00110001 119 11101101 170 11000111 221 00101011018 01100000 069 01100010 120 01110011 171 00100111 222 01010110019 11000000 070 11000100 121 11100110 172 01001110 223 10101100020 00101001 071 00100001 122 01100101 173 10011100 224 11110001021 01010010 072 01000010 123 11001010 174 10010001 225 01001011022 10100100 073 10000100 124 00111101 175 10001011 226 10010110023 11100001 074 10100001 125 01111010 176 10111111 227 10000101024 01101011 075 11101011 126 11110100 177 11010111 228 10100011025 11010110 076 01111111 127 01000001 178 00000111 229 11101111026 00000101 077 11111110 128 10000010 179 00001110 230 01110111027 00001010 078 01010101 129 10101101 180 00011100 231 11101110028 00010100 079 10101010 130 11110011 181 00111000 232 01110101029 00101000 080 11111101 131 01001111 182 01110000 233 11101010030 01010000 081 01010011 132 10011110 183 11100000 234 01111101031 10100000 082 10100110 133 10010101 184 01101001 235 11111010032 11101001 083 11100101 134 10000011 185 11010010 236 01011101033 01111011 084 01100011 135 10101111 186 00001101 237 10111010034 11110110 085 11000110 136 11110111 187 00011010 238 11011101035 01000101 086 00100101 137 01000111 188 00110100 239 00010011036 10001010 087 01001010 138 10001110 189 01101000 240 00100110037 10111101 088 10010100 139 10110101 190 11010000 241 01001100038 11010011 089 10000001 140 11000011 191 00001001 242 10011000039 00001111 090 10101011 141 00101111 192 00010010 243 10011001040 00011110 091 11111111 142 01011110 193 00100100 244 10011011041 00111100 092 01010111 143 10111100 194 01001000 245 10011111042 01111000 093 10101110 144 11010001 195 10010000 246 10010111043 11110000 094 11110101 145 00001011 196 10001001 247 10000111044 01001001 095 01000011 146 00010110 197 10111011 248 10100111045 10010010 096 10000110 147 00101100 198 11011111 249 11100111046 10001101 097 10100101 148 01011000 199 00010111 250 01100111047 10110011 098 11100011 149 10110000 200 00101110 251 11001110048 11001111 099 01101111 150 11001001 201 01011100 252 00110101049 00110111 100 11011110 151 00111011 202 10111000 253 01101010050 01101110 101 00010101 152 01110110 203 11011001 254 11010100__________________________________________________________________________ Note: S in polynomial notation has high order term on the left.
APPENDIX B
Theory for Decoding Two-Symbol Errors
This Appendix B provides the background for the decoding algorithm for two-symbol errors. This is derived from the well-known prior art method called Chien Search in decoding the generalized BCH code, which is described in the Chien paper cited above in the "Background Art" section.
Assume that there are exactly two erroneous bytes in one of the subblocks. The following proof will establish that error-location values y and z are two unique solutions of i in the equation:
T.sup.-2i P.sym.T.sup.-i Q=R (B-1)
where P, Q and R are functions of syndromes S.sub.0, S.sub.1, S.sub.2 and S.sub.3, as given by Equations (11)-(13). The error patterns E.sub.i for i=y or i=z each satisfies the following equation:
R=(T.sup.2i S.sub.0 .sym.S.sub.2) .circle..times. E.sub.i (B- 2)
Proof: The syndromes are expressed as functions of the two errors in Equations (7)-(10). These equations are rewritten here as:
S.sub.0 =E.sub.y .sym.E.sub.z (B- 3)
S.sub.1 =T.sup.y E.sub.y .sym.T.sup.z E.sub.z (B- 4)
S.sub.2 =T.sup.2y E.sub.y .sym.T.sup.2z E.sub.z (B- 5)
S.sub.3 =T.sup.3y E.sub.y .sym.T.sup.3z E.sub.z (B- 6)
Combining appropriate equations from (B-3) through (B-6), we have:
T.sub.yS0 .sym.S.sub.1 =(T.sup.y .sym.T.sup.z)E.sub.z (B- 7)
T.sup.y S.sub.1 .sym.S.sub.2 =T.sup.z (T.sup.y .sym.T.sup.z)E.sub.z (B- 8)
T.sup.y S.sub.2 .sym.S.sub.3 =T.sup.2z (T.sup.y .sym.T.sup.z)E.sub.z (B- 9)
Matrix Equations (B-7), (B-8), and (B-9) are relations among field elements in GF(2.sup.8) represented by matrices. In particular, the matrix multiplication of the type T.sup.i B represents the product of field element .alpha..sup.i and .beta., where .alpha..sup.i is represented by the first column of matrix T.sup.i, and .beta. is represented by the column vector B. In view of this interpretation, Equations (B-7), (B-8), and (B-9) yield the following relationship:
(T.sup.y S.sub.0 .sym.S.sub.1) .circle..times. (T.sup.y S.sub.2 .sym.S.sub.3)=(T.sup.y S.sub.1 .sym.S.sub.2).sup.2 (B- 10)
where .circle..times. denotes the product of corresponding elements in GF(2.sup.8). The Equation (B-10) can be rearranged into the following matrix equation:
T.sup.2y R.sym.T.sup.y Q.sym.P=0 (B-11)
In these equations, P, Q and R are column vectors given by:
P=(S.sub.2 .circle..times. S.sub.2).sym.(S.sub.3 .circle..times. S.sub.1) (B-12)
Q=(S.sub.2 .circle..times. S.sub.1).sym.(S.sub.3 .circle..times. S.sub.0) (B-13)
R=(S.sub.0 .circle..times. S.sub.2).sym.(S.sub.1 .circle..times. S.sub.1) (B-14)
Thus y is one of the solutions for i in the equation
T.sup.-2i P.sym.T.sup.-i Q=R (B-15)
By exchanging the variables y and z in the above process, it can be shown that z is the second solution for i in Equation (B-15).
Equation (B-2) for each error pattern can be verified by direct substitution of values for R, S.sub.0, S.sub.1 and S.sub.2. Both sides of
Equation (B-2) reduce to the expression:
(T.sup.2y .sym.T.sup.2z) (E.sub.y .circle..times. E.sub.z) (B-16)
thereby completing the proof.
APPENDIX C
Table Look-up Solution for Two Error Locations
In Appendix B, it was shown that the error locations y and z for two errors in a subblock can be determined by solving for i in Equation (B-1). That equation is rewritten here as:
T.sup.-2i P.sym.T.sup.-i Q=R (C-1)
The constants P, Q and R are functions of syndromes S.sub.0, S.sub.1, S.sub.2 and S.sub.3, given by Equations (B-12)-(B-14), respectively. We can obtain logarithms of P, Q and R from the log-antilog tables of Appendices A.1 and A.2.
p=log.sub..alpha. P (C-2)
q=log.sub..alpha. Q (C-3)
r=log.sub..alpha. R (C-4)
Then the matrix Equation (C-1) can be rewritten as a relation among field elements in GF(2.sup.8) as follows:
.alpha..sup.-2i. .alpha..sup.p .sym..alpha..sup.-i. .alpha..sup.q =.alpha..sup.r (C- 5)
Multiplying both sides of Equation (C-5) by .alpha..sup.p-2q, we obtain:
.alpha..sup.-2i+2p-2q) .sym..alpha..sup.(-i+p-q) =.alpha..sup.(r+p-2q) (C- 6)
Substituting t for (-i+p-q) in Equation (C-6), gives
.alpha..sup.2t .sym..alpha..sup.t =.alpha..sup.(r+p-2q) (C- 7)
and
i=(u)-t, (C-8)
where u=p-q
The right-hand side of Equation (C-7) is a known field element .alpha..sup.d, in which the exponent d is:
d=u+v, (C-9)
where u=p-q, and v=r-q
A table look-up solution is then provided for Equation (C-7) which can be rewritten as:
.alpha..sup.t (.alpha..sup.t .sym..alpha..sup.0)=.alpha..sup.d (C- 10)
Using this expression, each value of t (from 0 to 254) can be related to a value of d. Note that some values of d are absent in this relationship, and that each valid value of d corresponds to two values of t. For a given value of d, if t=t.sub.1 is one of the solutions of Equation (C-10), then it is apparent that t=t.sub.2 is also a solution where:
.alpha..sup.t.sbsp.2 =.alpha..sup.t.sbsp.1 .sym..alpha..sup.0 (C- 11)
Substituting t=t.sub.1 in Equation (C-10) and then using (C-11),
.alpha..sup.t.sbsp.1 .multidot..alpha..sup.t.sbsp.2 =.alpha..sup.d (C- 12)
Thus,
d=t.sub.1 +t.sub.2 (C- 13)
From equations (C-8), (C-9) and (C-13), the following two error location values i.sub.1 and i.sub.2 are obtained:
i.sub.1 =u-t.sub.1 (C- 14)
i.sub.2 =u-t.sub.2 =t.sub.1 -v (C-15)
Appendix C.1 relates each valid value of d to one of the two values of t. Values of d are listed in ascending order for easy reference as addresses of an 8.times.256-bit memory. The corresponding value of t is stored in memory as an 8-bit binary number. The all-zeros vector (invalid value shown by dashes in the table) is stored at addresses corresponding to the invalid values of d, and is so interpreted.
In case of two errors, the computed value of d fetches one of the two values for t from Appendix C.1. With this one value of t, Equations (C-14) and (C-15) provide the two values of i as the error locations y and z. An invalid value of d fetches t=0 from Appendix C.1, which is interpreted as an uncorrectable error involving three or more bytes of the codeword.
APPENDIX C.1__________________________________________________________________________Table of t vs d in: .alpha..sup.d = .alpha..sup.t (.alpha..sup.t +.alpha..sup.0)d t d t d t d t d t__________________________________________________________________________000 01010101 051 10001010 102 00010101 153 01000101 204 00101010001 -------- 052 -------- 103 10001011 154 10100101 205 11100101002 -------- 053 01001011 104 -------- 155 11001011 206 00010111003 00111000 054 -------- 105 00011111 156 10101011 207 --------004 -------- 055 10010111 106 10010110 157 00101110 208 --------005 -------- 056 00000100 107 10011100 158 00100001 209 --------006 01110000 057 01010111 108 -------- 159 -------- 210 00111110007 10000000 058 -------- 109 10010011 160 -------- 211 00100100008 -------- 059 01011100 110 00101111 161 -------- 212 00101101009 -------- 060 -------- 111 -------- 162 00110101 213 --------010 -------- 061 01000010 112 00001000 163 -------- 214 00111001011 00001100 062 -------- 113 -------- 164 -------- 215 --------012 00101011 063 -------- 114 10101110 165 00101001 216 --------013 -------- 064 -------- 115 01111001 166 00111101 217 11100010014 00000001 065 -------- 116 -------- 167 01001000 218 00100111015 -------- 066 -------- 117 -------- 168 01001101 219 --------016 -------- 067 -------- 118 10111000 169 01001111 220 01011110017 01110111 068 01100110 119 00010110 170 00100010 221 01011000018 -------- 069 01101010 120 -------- 171 -------- 222 --------019 01000001 070 -------- 121 -------- 172 00011011 223 00100110020 -------- 071 -------- 122 10000100 173 00111011 224 00010000021 01101011 072 -------- 123 -------- 174 -------- 225 --------022 00011000 073 -------- 124 -------- 175 -------- 226 --------023 -------- 074 -------- 125 -------- 176 11000000 227 --------024 01010110 075 01010010 126 -------- 177 -------- 228 01011101025 -------- 076 00000101 127 10011000 178 01000110 229 --------026 -------- 077 01111010 128 -------- 179 11000101 230 11110010027 -------- 078 01111000 129 00011100 180 00100101 231 --------028 00000010 079 10010000 130 -------- 181 01001110 232 --------029 -------- 080 -------- 131 01000000 182 11001001 233 00010010030 -------- 081 10011010 132 -------- 183 -------- 234 --------031 -------- 082 -------- 133 00000110 184 -------- 235 ---- ----032 -------- 083 10011110 134 -------- 185 10111100 236 01110001033 -------- 084 10100110 135 -------- 186 -------- 237 --------034 00110011 085 00010001 136 10111011 187 00001011 238 00101100035 -------- 086 10001101 137 10100000 188 -------- 239 00010011036 -------- 087 -------- 138 10110101 189 -------- 240 --------037 -------- 088 01100000 139 -------- 190 -------- 241 --------038 10000010 089 00100011 140 -- ------ 191 01001100 242 --------039 00111100 090 10010010 141 -------- 192 00001110 243 --------040 -------- 091 01110110 142 -------- 193 00100000 244 00001001041 -------- 092 -------- 143 -------- 194 00000011 245 --------042 01010011 093 -------- 144 -------- 195 -------- 246 --------043 01100100 094 -------- 145 -------- 196 01010000 247 01101110044 00110000 095 -------- 146 ------ -- 197 -------- 248 --------045 01001001 096 00000111 147 00011110 198 -------- 249 --------046 -------- 097 10000001 148 -------- 199 -------- 250 --------047 -------- 098 00101000 149 00110010 200 -------- 251 00110111048 10000011 099 -------- 150 10100100 201 00001111 252 --------049 00010100 100 -------- 151 -------- 202 00011001 253 01100010050 -------- 101 10001100 152 00001010 203 -------- 254 00110001__________________________________________________________________________ Note: Absence of a value for t implies an invalid value of d.

Number	Name	Date
4464753	Chen	Aug 1984
4525838	Patel	Jun 1985
4541092	Sako et al.	Sep 1985
4637021	Shenton	Jan 1987
4703485	Patel	Oct 1987
4706250	Patel	Nov 1987
4769818	Mortimer	Sep 1988

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