On the Extendibility of Near-MDS Elliptic Codes

The Main Conjecture on maximum distance separable (MDS) codes states that, except for some special cases, the maximum length of a q-ary linear MDS code of is q+1. This conjecture does not hold true for near maximum distance separable codes because of the existence of q-ary near-MDS elliptic codes having length bigger than q+1. An interesting related question is whether a near-MDS elliptic code may be extended to a longer near-MDS code. In this paper we prove some non-extendibility results for certain near-MDS elliptic codes.     

Note on Niederreiter-Xing's Propagation Rule for Linear Codes

We present a simple construction of long linear codes from shorter ones. Our approach is related to the product code construction; it generalizes and simplifies substantially the recent “Propagation Rule” by Niederreiter and Xing. Many optimal codes can be produced by our method. Received: June 23, 2000     

国外氨制冷系统中新的安全设置

本文介绍了国外在制冷系统中一些国内相关规范还没有的安全设置,以及这些安全设置的新工艺和新的做法.     

Convolutional Codes of Goppa Type

A new kind of Convolutional Codes generalizing Goppa Codes is proposed. This provides a systematic method for constructing convolutional codes with prefixed properties. In particular, examples of Maximum-Distance Separable (MDS) convolutional codes are obtained.This research was partially supported by the Spanish DGI through research project BFM2003-00078 and by the Junta de Castilla y León through research projects SA071/04 and SA032/02.     

Matrix-product codes over finite chain rings

Tim Blackmore and Graham H. Norton introduced the notion of matrix-product codes over finite fields. The present paper provides a generalization to finite chain rings. For codes a distance function is defined using a homogeneous weight function in the ring. It is proved that the minimum distance of a matrix-product codes is determined by the minimum distances of the separate codes. At the end of the paper we focus on Galois rings and define a special family of matrix-product codes.      

Convolutional Codes and Coherent Sheaves

We present a sheaf-theoretic setting for convolutional codes in that coherent sheaves (over a projective line) play the same role as finite-dimensional linear spaces in the theory of block codes. Using coherent sheaves (and their cohomologies), we give natural straightforward proofs of the main results on the structural properties of convolutional codes. Received: May 7, 1999; revised version: August 11, 2000     

Transform domain characterization of cyclic codes overZ m

Cyclic codes with symbols from a residue class integer ringZ _m are characterized in terms of the discrete Fourier transform (DFT) of codewords defined over an appropriate extension ring ofZ _m . It is shown that a cyclic code of length n overZ _m ,n relatively prime tom, consists ofn-tuples overZ _m having a specified set of DFT coefficients from the elements of an ideal of a subring of the extension ring. Whenm is equal to a product of distinct primes every cyclic code overZ _m has an idempotent generator and it is shown that the idempotent generators can be easily identified in the transform domain. The dual code pairs overZ _m are characterized in the transform domain for cyclic codes. Necessary and sufficient conditions for the existence of self-dual codes overZ _m are obtained and nonexistence of self-dual codes for certain values ofm is proved.     

Computation of the weight distribution of CRC codes

In this article, we illustrate an algorithm for the computation of the weight distribution of CRC codes. The recursive structure of CRC codes will give us an iterative way to compute the weight distribution of their dual codes starting from some “representative” words. Thanks to MacWilliams’ Theorem, the computation of the weight distribution of the dual codes can be easily brought back to that of CRC codes. The author was partially supported by the Swiss National Science Foundation under Grant no. 113251.     

Linear codes with balanced weight distribution

In this paper, we study particular linear codes defined overF _q , with an astonishing property, their weight distribution is balanced, i.e. there is the same number of codewords for each nonzero weight of the code. We call these codesBWD-codes. We first study BWD-codes by means of the Pless identities and we completely characterize the two-weight projective case. We study the class of codes defined under subgroups of the multiplicative group ofF _q ^s , using the Gauss sums. Then, given a primep and an integerN dividingp – 1, we construct all theN-weight BWD-codes of that class. We conclude this paper by some tables of BWD-codes and an open problem.     

Minimum distance of relative Reed–Muller codes

We describe a construction of error-correcting codes on a fibration over a curve defined over a finite field, which may be considered as a relative version of the classical Reed–Muller code. In the case of complete intersections in a projective bundle, we give an explicit lower bound for the minimum distance.     

Error-correcting codes and cryptography

In this paper, we give and explain some illustrative examples of research topics where error-correcting codes overlap with cryptography. In some of these examples, error-correcting codes employed in the implementation of secure cryptographic protocols. In the others, the codes are used in attacks against cryptographic schemes. Throughout this paper, we show the interrelation between error-correcting codes and cryptography, as well as point out the common features and the differences between these two fields.     

Weight distribution and dual distance

Several results in coding theory (e.g. the Carlitz-Uchiyama bound) show that the weight distributions of certain algebraic codes of lengthn are concentrated aroundn/2 within a range of width n. It is proved in this article that the extreme weights of a linear binary code of sufficiently high dual distance cannot be too close ton/2, the gap being of order n. The tools used involve the Pless identities and the orthogonality properties of Krawtchouk polynomials, as well as estimates on their zeroes. As a by-product upper bounds on the minimum distance of self-dual binary codes are derived.     

Improved Bounds for Quaternary Linear Codes of Dimension 6

In this paper, twenty new codes of dimension 6 are presented which give improved bounds on the maximum possible minimum distance of quaternary linear codes. These codes belong to the class of quasi-twisted (QT) codes, and have been constructed using a stochastic optimization algorithm, tabu search. A table of upper and lower bounds for d ⁴(n,6) is presented for n≤ 200. Received: 20 December 1996 / Accepted: 13 May 1997     

On New Perfect Binary Nonlinear Codes

 We present a new construction of binary nonlinear perfect codes with minimum distance 3 and lowerbound the number of nonequivalent such codes. Received: November 26, 1996; revised version: March 14, 1997     

On the Equivalence of Codes over Finite Rings

It is known that if a finite ring R is Frobenius then equivalences of linear codes over R are always monomial transformations. Among other results, in this paper we show that the converse of this result holds for finite local and homogeneous semilocal rings. Namely, it is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a codeC₁ onto a code C₂ is a monomial transformation then R is a Frobenius ring.Partially supported by Center of Ring Theory and Applications, Ohio University.     

Extended Classical Goppa Codes

We construct extended classical Goppa codes that can have unrestricted block length. The parameters of the codes are estimated, and the standard Berlekamp–Massey error processor is adapted to the codes. Received: July 5, 1999; revised version: February 25, 2000     

Construction of Improved Geometric Goppa Codes from Klein Curves and Klein-like Curves

Linear error-correcting codes, especially Reed-Solomon codes, find applications in communication and computer memory systems, to enhance their reliability and data integrity. In this paper, we present Improved Geometric Goppa (IGG) codes, a new class of error-correcting codes, based on the principles of algebraic-geometry. We also give a reasonably low complexity procedure for the construction of these IGG codes from Klein curves and Klein-like curves, in plane and high-dimensional spaces. These codes have good code parameters like minimum distance rate and information rate, and have the potential to replace the conventional Reed-Solomon codes in most practical applications. Based on the approach discussed in this paper, it might be possible to construct a class of codes whose performance exceeds the Gilbert-Varshamov bound. Received: November 14, 1995; revised version: November 22, 1999     

On the Covering Radius of Long 5-ary BCH Codes with Minimum Distance 7

 We study S. D. Cohen’s method of estimating the covering radius of long primitive non-binary BCH codes with prime field alphabet. As an application we show that the covering radius is at most 7 for long 5-ary BCH codes with minimum distance seven. Received: November 13, 1996; revised version: March 11, 1997     

Graded Codes

 Since S.D. Berman showed in 1967 that cyclic codes and Reed Muller codes can be studied as ideals in a group algebra KG (over a finite field K and G is considered, in each case, a finite cyclic group and a 2-group respectively), several authors have investigated these codes. It has been observed, that the presence of additional algebraic structure make their study more effective. Following these ideas, we consider codes which are (graded) left ideals of a graded ring R or more generally, codes as (graded) left R-modules. In this paper, a linear code is graded if it is a graded R-module for some multiplicative group G and a G-graded K-algebra R. We will show that some important properties of graded codes can be obtained from their homogeneous components and then we generalize some results about codes as ideals in group algebras. In particular, we study cocyclic codes as ideals in a twisted group algebra, and more generally, in a crossed product. Received: May 22, 1999; revised version: August 20, 2001     

Reed-Muller Codes on Complete Intersections

By using results and techniques from commutative algebra such as the vanishing ideal of a set of points, its a-invariant, the Hilbert polynomial and series, as well as finite free resolutions and the canonical module, some results about Reed-Muller codes defined on a zero-dimensional complete intersection in the n-projective dimensional space are given. Several examples of this class of codes are presented in order to illustrate the ideas. Received: March 11, 1999; revised version: November 6, 2000