Decoding cyclic and BCH codes up to actual minimum distance using nonrecurrent syndrome dependence relations

The decoding capabilities of algebraic algorithms, mainly the Berlekamp-Massey algorithm, the Euclidean algorithm and our generalizations of these algorithms, are basically constrained by the minimum distance bounds of the codes. Thus, when the actual minimum distance of the codes is greater than that given by the bounds, these algorithms usually cannot fully utilize the error-correcting capability of the codes. The limitation is seen to be rooted in the original Peterson decoding procedure adhered to by these algorithms. Thus, these algorithms all require the determination of the error-locator polynomial from Newton’s identities which in turn require that the syndromes be contiguous in forming a set or multiple sets of linear recurrences. A procedure is introduced that breaks away from this restriction and can determine the error locations from nonrecurrent syndrome dependence relations. This procedure employs an algorithm that has recently been introduced as a basis for the derivation of the Berlekamp-Massey algorithm and its generalization. It can decode many cyclic and BCH codes up to their actual minimum distance and is seen to he a generalization of Peterson’s procedure. Index Terms -Cyclic coding, BCH coding, generalization of Peterson decoding procedure, decoding up to actual minimum distance. I. INTRODUC~ION Algorithms for algebraic decoding of cyclic and BCH codes, mainly the Berlekamp-Massey algorithm [l], 121, the Euclidean algorithm [3], as well as our generalizations of these algorithms [4], [SI basically suffer from a restriction imposed by the miniManuscript received July 24, 1990; revised May 7, 1991. This work was supported by the National Science Foundation under Grant NCR8716953. This work was presented in part at the 1990 IEEE International Symposium on Information Theory, San Diego, CA, January G.-L. Feng was with the Department of Computer Science and Electrical Engineering, Lehigh University, Bethlehem, PA 18015. He is now with the Center for Advanced Computer Studies, University of Southwestern Louisiana, Lafayette, LA 70504. K. K. Tzeng is with the Department of Computer Science and Electrical Engineering, Lehigh University, Bethlehem, PA 18015. IEEE Log Number 9102344. 14-19, 1990. mum distance bounds of the codes. For the Berlekamp-Massey algorithm and the Euclidean algorithm, this restriction comes from the BCH bound as these algorithms can normally decode only up to this bound. Likewise, our generalizations of these algorithms usually cannot decode beyond the Hartmann-Tzeng (HT) bound and the Roos bound [4], [6]-[8]. Several authors [9]-[15] have attempted to stretch the capability of the Berlekamp-Massey algorithm for decoding beyond the BCH bound and have succeeded in various degrees for particular cases. But, generally speaking, when the actual minimum distance of the codes is greater than that given by such bounds, these algebraic algorithms usually are not able to utilize the full error-correcting capability of the codes. The limitations are seen to be originated in the Peterson procedure for decoding BCH codes [16] adhered to by these algorithms. As such, these algorithms all require the determination of the error-locator polynomial from Newton’s identities which in turn require that the syndromes be contiguous in forming a set or multiple sets of linear recurrence relations. This, of course, is a consequence of the contiguity required on the roots of the generator polynomial by these bounds. In this correspondence, we introduce a more general procedure which breaks away from this restriction imposed by the minimum distance bounds and can determine the error locations from nonrecurrent dependence relations among the syndromes. This procedure employs an algorithm, referred to as the Fundamental Iterative Algorithm, which we have recently introduced as a basis for the derivation of the Berlekamp-Massey algorithm and its generalization [4]. The procedure can decode many cyclic and BCH codes up to their actual minimum distance and is seen to be a generalization of Peterson’s procedure. 11. PRELIMINARIES In this section, we give a brief review of the Peterson decoding procedure and the Fundamental Iterative Algorithm for ease of later reference. Let g(x) be the generator polynomial of a cyclic code of length n over GF(q) and let d be the actual minimum distance of this code. The code is then capable of correcting up to t = [ ( d 11/21 errors. Let e(x) = CL=,Y,Xau with v I l , 0 I a , < a 2 < . . . < a v < n and Y,#O for p = 1 , 2 ; . . , v , be an error polynomial resulted from some transmitted code polynomial ~ ( x ) . Then the received polynomial is r ( x ) = u(x>+ e h ) . Suppose, for some primitive nth root of unity p E GF(q”), g ( p k ) = 0. Then u ( p k > = 0 and r ( p k ) = e@,>. Thus the syndrome term S, = can be computed from the received polynomial. Furthermore, we have Sqk = S,4 and S, + k = s k . For BCH codes, and cyclic codes in general, d o 1 “consecutive” syndrome terms are known where do denotes the BCH bound. Suppose g ( p b + ” ) = 0 where b is any integer, c is an integer relatively prime to n and i = 0,l; . . ,do -2. Then S b , S b + c , . ’ . , S b + ( d o 2 ) c are known, where s ~ + , ~ = e( P’+~“) with X , = pa@ and i = 0,1,. . . , d o 2. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 6, NOVEMBER 1991 1717 The problem of decoding BCH codes is to determine the error locations X,’s and the error magnitudes Yw’s from the d o 1 syndromes. The procedure devised by Peterson and generalized to the nonbinary cases by Gorenstein and Zierler [16] is to separate the issues of determining the error locations and the error magnitudes by first defining the error-locator polynomial developed in [4]. A brief review of this algorithm is given in the following. + a j , l ~ + . . . + u , , ~ x ~ , where co = 1 and = 1, for i = 1,2; . ., M . For I +1 I n I N , let [C(x)a(’)(x)], be the coefficient of X” in C(x)a( ‘ ) (x) , namely Let C(x) = co +‘c,x + . . . + cIx‘ and a( ’ ) (x) = U a@)= n (2 -x i ) p = 1 = Z Y + a1zy-1 + . . . + avy-12 +a,. Then, These recurrence relations have been referred to as the generalized Newton identities. The problem is now transformed to the determination of a ( z ) from the d o 1 syndromes through (2). After a ( z ) is determined, the error locations will be given by the roots of a ( z ) . Then the error magnitudes can be determined easily. The procedure thus consists of the following steps: 1) calculate the syndromes S , , 2) determine d z ) from (21, 3 ) determine the error locations X,, 4) determine the error magnitudes Y,. The second step is now best accomplished by the Berlekamp-Massey algorithm. The third step can be handled efficiently by the Chien search and the last step is completed by using Forney’s formula [16]. Alternatively, after the a,’s are determined, the unknown syndromes can be determined through (2). When So, SI,. . , S, all become known, then, as shown by Blahut [19], an inverse Fourier transform will determine all the error locations and all the error magnitudes. However, this procedure can only decode up to t = l (do 11/21 errors. Similarly, when the lower bound on the minimum distance of the code is given by the HT bound d, , or the Roos bound dRoos, the generalized algorithms can only decode up to Id,, 1/21 or [dRoos 1/21 errors [4], [8]. To make it clear, we shall also use d,,, to denote the BCH bound. In the next section, we shall present a more general procedure for decoding up to the actual minimum distance. The main feature of this procedure is the incorporation of the fundamental iterative algorithm in determining the error locator polynomial from a nonrecurrent syndrome dependence relation. This algorithm is for finding the smallest initial set of dependent columns in an M X N matrix over any field F with rank less than N. Let a11 a12 . . . a l N . . . . . .