Kenneth K. Tzeng

A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes

A generalization of the Berlekamp-Massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences. A more general problem is first considered, that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field, which includes the multisequence problem as a special case. A simple iterative algorithm, the fundamental iterative algorithm (FIA), is presented for solving this problem. The generalized algorithm is then derived through a refinement of the FIA. Application of this generalized algorithm to decoding cyclic codes up to the Hartmann-Tzeng (HT) bound and Roos bound making use of multiple syndrome sequences is considered. Conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error-locator polynomial are determined with respect to decoding up to the HT bound and special cases of the Roos bound. >

Generalizations of the BCH bound

Cyclic codes generated by polynomials having multiple sets of do — 1 roots in consecutive powers of a nonzero field element are considered and some generalizations of the BCH bound are presented. In particular, it is shown, among other results, that if g(x) GF(q)[x[ is the generator polynomial of a cyclic code Vn of length n such that g(βl+i1c1+i2c2 = 0 for i1 = 0, 1,…, d0 — 2 and i2 = 0,1,…,s, where β GF(qm) is a nonzero element of order n and c1 , c2 are relatively prime to n, then the minimum distance of Vn is at least d0 + s.

A generalized Euclidean algorithm for multisequence shift-register synthesis

The problem of finding a linear-feedback shift register of shortest length capable of generating prescribed multiple sequences is considered. A generalized Euclidean algorithm, which is based on a generalized polynomial division algorithm, is presented. A necessary and sufficient condition for the uniqueness of the solution is given. When the solution is not unique, the set of all possible solutions is also derived. It is shown that the algorithm can be applied to the decoding of many cyclic codes for which multiple syndrome sequences are available. When it is applied to the case of a single sequence, the algorithm reduces to that introduced by Y. Sugiyama et al. (Inf. Control, vol.27, p.87-9, Feb. 1975) in the decoding of BCH codes. >

On the generalized Hamming weights of several classes of cyclic cods

The generalized Hamming weights of a linear code are fundamental code parameters related to the minimal overlap structures of the subcodes. They were introduced by V.K. Wei (1991) and shown to characterize the performance of the linear code in certain cryptographical applications. Results are presented on the generalized Hamming weights of several classes of binary cyclic codes, including primitive double-error-correcting and triple-error-correcting BCH codes, certain reversible cyclic codes, and some extended binary Goppa codes. In particular, the second generalized Hamming weight of primitive double-error-correcting BCH codes is determined and upper and lower bounds are obtained for the generalized Hamming weights for the codes studied. These bounds are compared to results from other methods. >

Simplified understanding and efficient decoding of a class of algebraic-geometric codes

An efficient decoding algorithm for algebraic-geometric codes is presented. For codes from a large class of irreducible plane curves, including Hermitian curves, it can correct up to [(d*-1)/2] errors, where d* is the designed minimum distance. With it we also obtain a proof of d/sub min//spl ges/d* without directly using the Riemann-Roch theorem. The algorithm consists of Gaussian elimination on a specially arranged syndrome matrix, followed by a novel majority voting scheme. A fast implementation incorporating block Hankel matrix techniques is obtained whose worst-case running time is O(mn/sup 2/), where m is the degree of the curve. Applications of our techniques to decoding other algebraic-geometric codes, to decoding BCH codes to actual minimum distance, and to two-dimensional shift register synthesis are also presented. >

A new procedure for decoding cyclic and BCH codes up to actual minimum distance

The paper presents a new procedure for decoding cyclic and BCH codes up to their actual minimum distance. It generalizes the Peterson decoding procedure and the procedure of Feng and Tzeng (1991) using nonrecurrent syndrome dependence relations. For a code with actual minimum distance d to correct up to t=[(d-1)/2] errors, the procedure requires a (2t+1)/spl times/(2t+1) syndrome matrix with known syndromes above the minor diagonal and unknown syndromes and their conjugates on the minor diagonal. In contrast to previous procedures, this procedure is primarily aimed at solving for the unknown syndromes instead of determining an error-locator polynomial. Decoding is then accomplished by determining the error vector as the inverse Fourier transform of the syndrome vector (S/sub 0/, S/sub 1/, S/sub n-1/). The authors show that with this procedure, all binary cyclic and BCH codes of length >

On extending Goppa codes to cyclic codes (Corresp.)

The class of linear codes introduced by Goppa are noncyclic in general. The only Goppa codes known to be cyclic are the Bose--Chaudhuri-Hocquengbem (BCH) codes which are a special class of Goppa codes. Recently, Berlekamp and Moreno showed that certain double-error-correcting binary Goppa codes become cyclic when extended by an overall parity check. In this correspondence, results of a further investigation on extending Goppa codes to cyclic codes are presented. It is shown that a large class of multiple-error-correcting q -ary Goppa codes also become cyclic when extended by an overall parity check. These Goppa codes are found to be reversible. The Goppa codes considered in this correspondence consist of two subclasses that, after extension, give rise to the two subclasses of reversible cyclic codes of primitive and nonprimitive length, respectively. These cyclic codes are noted to include, respectively, the expurgated Melas codes and the Zetterberg codes as special cases.

Some results on the minimum distance structure of cyclic codes

This paper presents a number of interesting results relating to the determination of actual minimum distance of cyclic codes. Codes with multiple sets of consecutive roots are constructed. A bound on the minimum weight of odd-weight codewords is determined. Relations on the distribution of roots of the generator polynomial are investigated. Location polynomials of reversible codes are examined. These results are used to obtain better estimates of the minimum distance of many new cyclic codes.

On the minimum distance of certain reversible cyclic codes (Corresp.)

The minimum distance of a class of reversible cyclic codes has been proved to be greater than that given by the BCH bound. It is also noted that this class of codes includes the class of primitive double-error-correcting binary codes of Melas as well as the class of nonprimitive double-error-correcting binary codes discovered by Zetterberg as special cases.

Decoding beyond the BCH bound using multiple sets of syndrome sequences (Corresp.)

Many cyclic codes are generated by polynomials possessing more than one set of consecutive roots. Thus more than one set of syndrome sequences are available for decoding. In this correspondence, a decoding method based on Berlekamps iterative algorithm is presented which makes use of the multiple sets of syndrome sequences for decoding such cyclic codes beyond the BCH bound.