Maximum-likelihood soft decision decoding of BCH codes

The problem of efficient maximum-likelihood soft decision decoding of binary BCH codes is considered. It is known that those primitive BCH codes whose designed distance is one less than a power of two, contain subcodes of high dimension which consist of a direct sum of several identical codes. We show that the same kind of direct-sum structure exists in all the primitive BCH codes. as well as in the BCH codes of composite block length. We also introduce a related structure termed the “concurring-sum”, and then establish its existence in the primitive binary BCH codes. Both structures are employed to upper bound the number of states in the proper minimal trellis of BCH codes, and develop efficient algorithms for maximum-likelihood soft decision decoding of these codes. In [2] Forney has shown that the binary Reed-Muller codes contain directsum subcodes of high dimension. It is well known that certain BCH codes, namely the primitive binary BCH codes with designed distance one less than a power of two, are supercodes of punctured Reed-Muller codes. Hence these BCH codes evidently share the direct-sum structure of the RM codes. This fact was used by Kasami et al. [3] to construct efficient trellis diagrams for the (64,24,16) and (64,45,8) extended BCH codes, and also several doubleerror correcting BCH codes. The following question, hence, arises: do other BCH codes also contain direct-sum subcodes of high dimension? We settle this question affirmatively for all the primitive BCH codes, and also for the BCH codes of composite block length. The direct-sum structure is in a sense a counterpart of the concept of “zero-concurring” codewords of [I, 41, obtained by substituting a code for each codeword. We also study a different structure, where we allow the constituent codes to overlap over a fixed set of coordinates. This concumng-sum structure is the corresponding counterpart of the “concurring” codewords of [I]. We show the existence of concurring-sum structures in all the primitive binary BCH codes. Both the directand the concurring-sum structures make it possible to set nontrivial upper bounds on the number of states in the minimal proper trellis of BCH codes, and provide a clue for efficient soft-decision decoding. Let C be a binary BCH code of length n and dimension I C , let a be a primitive nth root of unity, and let Z be a subset of (0, 1, . . . n-1). Denote by C[z] the subcode of C which consists of all those codewords that are nonzero only on the positions contained in 2. Let C(Z) be the code obtained from C[q by puncturing out all the positions not in Z. Proposition 1. Let 21 and Zz be subsets of the set (0, 1, . . . n-l}, such that for some a E {O,l,. . .n-1) we have {U’ : i E 12) = {a’.a‘ : i E 11). Then Now assume that the block length of C is composite, say n = nl‘nz, and let Z be the set of zero frequencies of C. Define S = ( 8 3 z (mod nl) : z E Z}. Proposition 2. LetZi = {O,nz,2n2,. . . (nl-l)nz). Then the code C(Z1) is a BCH code of length n1 and drmenszon kl = nl JSJ. The zeros of C(Z,) lie at {p” : s E S} , where ,O = a n a zs a pnnztzve nih root of unrty. In order to obtain direct-sum subcodes of high dimension in BCH codes of composite block length, it would now suffice to partition the set (0, 1, . . n-1) into nz disjoint subsets satisfying the condition of Proposition 1 with respect to the set Z1 defined in PropositionZ. Note that the sets Z and S are unions of cyclotomic cosets modulo n and nl, respectively. Thus the definition of S in conjunction with Proposition 2 induces “coset aliasing” between the cyclotomic cosets modulo n and modulo nl. In particular, certain high frequencies of C alias as low frequencies in C(Z1). This is intuitively plausible since 1 1 is just the “time-domain sampling” of C. In the sequel we consider the primitive BCH codes. Henceforth let C denote an extended primitive narrow-sense BCH code of length n + 1 = Zm. Proposition 3. Let Z1 and Zz be subsets of the set {0,1,. . . n-I, CO}, such that for some a E {0,1,. . .n-1) we have {aa : i E 2 2 ) = {a“ +U‘ : i E 11). Then C(Z1) = C(Zz). Proposition3 may be thought of as the “addition counterpart” of Proposition 1. Thus we can exhibit the existence of direct-sum subcodes in the extended primitive BCH codes by partitioning the set {ao,al , . . ~ “ ~ , a ~ } into disjoint subsets satisfying the condition of Proposition3 with respect to some given subset. Yet this set is just the field GF(2m). Thus it would suffice to regard GF(2m) as a vector space, and partition it into a subspace and its cosets. Notably, Proposition3 may be also employed for the derivation of the concurring-sum structure in the primitive binary BCH codes. For more details on this see [5]. . C(Z1) = e(&).