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Note that all these columns are of odd weight. 2.3.5 Binary BCH Codes
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BCH codes are a class of multiple random error correcting codes [BOSE60, HOCQ59]. In this subsection we cover binary BCH codes that are used to correct small numbers of random errors. For a detailed description of BCH codes, the reader is referred to the texts on coding theory. Among the nonbinary BCH codes, the next subsection covers the most important class of Reed-Solomon (RS) codes. De nition 2.27 BCH codes are cyclic codes generated by g x involving multiple factors. For a t-error correcting binary BCH code, the generator polynomial is given by g x m1 x m3 x m2t 1 x ; where mi x is the minimal polynomial of ai ; i 1; 3; 5; . . . ; 2t 1, and a is an element of GF 2m of order n. In this case, if m1 x is a primitive polynomial of degree m over GF 2 , the code is called a primitive binary BCH code. Then a is a primitive element & and its order n 2m 1. In the primitive binary BCH codes, the roots of the factors of g x are as follows: m1 x : m3 x : m5 x : m2t 1 x : a; a2 ; a4 ; ; a3 ; a6 ; ; a5 ; a10 ; ; a2t 1 ; a4t 2 ; :
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The 2t successive powers of a (i.e., a; a2 ; a3 ; ; a2t 1 ; a2t ) are roots of g x . A polynomial v x is a code polynomial if and only if v aj 0 for j 1; 2; ; 2t. That is, this means v HT 0, where H is a parity-check matrix of the BCH code that corrects the random t errors as follows: 1 61 6 6 H 61 6. 4. . 1 2 a a2 a3 . . . a2t a2 a2 2 a3 2 . . . a2t 2 an 1 a2 n 1 a3 n 1 . . . 3 7 7 7 7: 7 5 2:15
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In Eq. (2.15) the transmitted codeword is expressed as V v0 ; v1 ; v2 ; . . . ; vn 1 . In order to prove that the code has a distance at least 2t 1, we need to show that every 2t column vectors of H are linearly independent. By choosing any 2t columns from the matrix, we have a 2t 2t square matrix. After normalizing every column by the rst row element, we have the Vandermonde matrix (shown in Subsection 2.2.3) with a nonzero determinant. This means that every 2t columns of H are linearly independent. i If a is a root of g x over GF 2 , then its conjugate a2 is also a root, which is shown in Subsection 2.1.4. For this reason even rows can be omitted from matrix (2.15). As a result the H matrix can be reduced to the following: 1 61 6 6 H 61 6. 4. . 1 2 a a3 a5 . . . a2t 1 a2 a3 2 a5 2 . . . a2t 1 2 an 1 a3 n 1 a5 n 1 . . . 3 7 7 7 7: 7 5 2:16
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Note here that elements of H are in GF 2m , so each element can be represented by an m-tuple over GF 2 . That is, the total number of check bits is tm. The binary BCH code has the following code parameters: Maximum code length in bits: n 2m 1, Check-bit length: n k tm, Minimum distance: dmin ! 2t 1. For example, using Table 2.3, we can design the triple-bit error correcting BCH code by choosing three minimal polynomials corresponding to a, a3 , and a5 as their roots over GF 24 . That is, the generator polynomial is expressed as g x x4 x 1 x4 x3 x2 x 1 x2 x 1 : The code has 10 check bits, and these are at maximum 15 bits in the code length. The primitive binary BCH code is de ned by the generator polynomial g x m1 x , where m1 x is a binary primitive polynomial with degree m, and it expresses a singleerror correcting BCH code with code length n 2m 1. This is same as the Hamming SEC code described in the previous subsection on cyclic codes.
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