Gui Liang Feng

Decoding algebraic-geometric codes up to the designed minimum distance

A simple decoding procedure for algebraic-geometric codes C/sub Omega /(D,G) is presented. This decoding procedure is a generalization of Petersons decoding procedure for the BCH codes. It can be used to correct any ((d*-1)/2) or fewer errors with complexity O(n/sup 3/), where d* is the designed minimum distance of the algebraic-geometric code and n is the codelength. >

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. >

Improved geometric Goppa codes. I. Basic theory

In this paper, we present a construction of improved geometric Goppa codes which, for the case of r<2g, are often more efficient than the current geometric Goppa codes derived from some varieties, which include algebraic curves, hyperplanes, surfaces, and other varieties. For the special case of a plane in a three-dimensional projective space, the improved geometric Goppa codes are reduced to linear multilevel codes. For these improved geometric Goppa codes, a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given.

A simple approach for construction of algebraic-geometric codes from affine plane curves

The current algebraic geometric (AG) codes are based on the theory of algebraic geometric curves. In this paper, we present a novel approach for construction of AG codes without any background in algebraic geometry. Given an affine plane irreducible curve and its all rational points, based on the equation of this curve, we can find a sequence of monomial polynomials x/sup i/y/sup j/. Using the first r polynomials as a basis of dual code of a linear code called AG code, the designed minimum distance d of this AG code can be easily determined. For these codes a fast decoding procedure with complexity O(n/sup 7/3/) which can correct errors up to [(d-1)/2], is also shown. By this approach it is neither necessary to know the genus of curve nor find a basis of differential form. This approach can be easily understood by most engineers. Some examples are also shown, which indicate that the codes constructed by this approach are better than the current AG codes from same curves.

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 >

Improved lower bounds on the sizes of error-correcting codes for list decoding

Elias (1991) derived upper and lower bounds on the sizes of error-correcting codes for list decoding. The asymptotic values of his lower bounds for linear codes and for nonlinear codes are separated. The present authors derive improved lower bounds for linear and for nonlinear codes. They conjecture their two bounds are identical. However, they were able to verify this only for small lists. >

On quasi-perfect property of double-error-correcting Goppa codes and their complete decoding

In this paper, it is proved that the binary Goppa codes with L = GF (2 m ) and G ( z ) = z 2 + z + β are quasi-perfect when m is odd and are nearly-quasi-perfect when m is even. In particular, it is shown that for any syndrome, except the case where m is even and the syndrome terms are s 1 = 0 and s 3 = 1, the corresponding coset is of weight ⩽3. For the exceptional case, it is shown that the corresponding coset is of weight 4. The results thus complement to a large extent those previously reported by Moreno. Furthermore, the proofs given are constructive and offer a method of complete decoding of such Goppa codes.