Ternary Covering Codes Derived from BCH Codes

It is shown how ternary BCH codes can be lengthened to get linear codes with covering radius 2. The family obtained has the ternary Golay code as its first code, contains codes with record-breaking parameters, and has a good asymptotic behavior. The ternary Golay code is further used to obtain short proofs for the best known upper bounds for the football pool problem for 11 and 12 matches.     

The Length of Primitive BCH Codes with Minimal Covering Radius

It is proved that the covering radius of a primitive binary BCH code of length q-1 and designed distance 2t+1, where is exactly 2t-1 (the minimum value possible). The bound for q is significantly lower than the one obtained by O. Moreno and C. J. Moreno [9].     

On the Distance Distributions of BCH Codes and Their Duals

Based on new bounds on the values of Krawtchoukpolynomials, we improve earlier known estimates forcomponents of the distance distributions of BCH codes and theirduals. Moreover, we show that if one uses estimates on Krawtchoukpolynomials for bounding the error term in the binomial approximationto the distance distribution of BCH codes, the given results are actuallythe best possible. One of the advantages of the proposed approach isthat it provides estimates with no restrictions on the minimumdistance of the code.     

The Automorphism Groups of BCH Codes and of Some Affine-Invariant Codes Over Extension Fields

Affine-invariant codes are extended cyclic codes of length p ^m invariant under the affine-group acting on . This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.     

Hypercubic 4 and 5-Designs from Double-Error-Correcting BCH Codes

An ordered orthogonal array OOA(, k, n) is a binary 2 ^k × n matrix with the property that for each complete -set of columns, each possible -tuple occurs in exactly 2 ^k– rows of those columns (for definition of a complete -set, see below). Constructions of OOA(, k, n) for = 4 and = 5 are given.     

关于设计距离为7的BCH码的非循环等价类

利用Mobius公式给出了设计距离为7的二元BCH码的非循环等价类的精确计数．     

An Algorithm for Computing the Minimum Distances of Extensions of BCH Codes Embedded in Semigroup Rings

An algorithm is given for computing the weights of extensions of BCH codes embedded in semigroup rings as ideals. The algorithm relies on a more general technical result of independent interest.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

New Constructions of Orbit Codes Based on the Operations of Orbit Codes

An orbit code is a special constant dimension subspace code, which is an orbit of a subgroup of a general linear group acting on the set of all subspaces in the given ambient space. This paper presents some methods of constructing new orbit codes from known orbit codes. Firstly, we introduce the sum operation,intersection operation and union operation of subspace codes, and then we give some methods to obtain new orbit codes from known orbit codes by fully applying the sub-orbits of permutation ...     

New Quaternary Linear Codes with Covering Radius 2

A new quaternary linear code of length 19, codimension 5, and covering radius 2 is found in a computer search using tabu search, a local search heuristic. Starting from this code, which has some useful partitioning properties, different lengthening constructions are applied to get an infinite family of new, record-breaking quaternary codes of covering radius 2 and odd codimension. An algebraic construction of covering codes over alphabets of even characteristic is also given.     

Existence of New Extremal Doubly-Even Codes and Extremal Singly-Even Codes

Recently the author and Kimura have considered a construction of doubly-even codes from a given doubly-even code. In this note, we show that the restricutoion of doubly-even can be removed in the above construction. As an application, at least 137 inequivalent extremal doubly-even [56,28,12] codes and at least 1000 inequivalent extremal doubly-even [40,20,8] codes are constructed from known self-dual codes. The existence of new extremal singly-even codes is also described.     

New Constructions of Covering Codes

Coveringcode constructions obtaining new codes from starting ones weredeveloped during last years. In this work we propose new constructionsof such kind. New linear and nonlinear covering codes and aninfinite families of those are obtained with the help of constructionsproposed. A table of new upper bounds on the length functionis given.     

Codes with the Same Weight Distributions as the Goethals Codes and the Delsarte-Goethals Codes

The Goethals code is a binary nonlinear code of length 2^m+1 which has codewords and minimum Hamming distance 8 for any odd . Recently, Hammons et. al. showed that codes with the same weight distribution can be obtained via the Gray map from a linear code over Z ₄of length 2^m and Lee distance 8. The Gray map of the dual of the corresponding Z ₄ code is a Delsarte-Goethals code. We construct codes over Z ₄ such that their Gray maps lead to codes with the same weight distribution as the Goethals codes and the Delsarte-Goethals codes.     

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.     

New Constructions for IPP Codes

Identifiable parent property (IPP) codes are introduced to provide protection against illegal producing of copyrighted digital material. In this paper we consider explicit construction methods for IPP codes by means of recursion techniques. The first method directly constructs IPP codes, whereas the second constructs perfect hash families that are then used to derive IPP codes. In fact, the first construction provides an infinite class of IPP codes having the best known asymptotic behavior. We also prove that this class has a traitor tracing algorithm with a runtime of O(M) in general, where M is the number of codewords.     

Weight Enumerators of Double Circulant Codes and New Extremal Self-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.     

Two New Four-Error-Correcting Binary Codes

Two four-error-correcting binary codes of length 21 and 22 and of cardinality 64 and 80, respectively, are constructed. The codes consist of a union of cosets of linear codes with dimension 3 and were found by a maximum clique algorithm.     

叠加码的一些新构作

利用集合的并与直积运算建立了新的二元矩阵,从而在已有的叠加码的基础上构作了一些新的叠加码.