New Codes via the Lengthening of BCH Codes with UEP Codes

We describe a code lengthening technique that uses unequal error protection codes as suffix codes and combine it with iteration of the conventional Construction X. By applying this technique to BCH codes, we obtain five new binary codes, 13 new ternary codes, and 13 new quarternary codes. An improvement of Construction XX yields two new ternary codes.     

Asymptotically Good Codes from Generalized Algebraic-Geometry Codes

We consider generalized algebraic-geometry codes, based on places of the same degree of a fixed algebraic function field over a finite field. In this note, using a method similar to the Justesen’s one, we construct a family of such codes which is asymptotically good.Communicated by: D. Jangnickel     

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.     

模糊化内缀码和模糊化外缀码

在自由幺半群上引进模糊化内缀码和模糊化外缀码的概念,并进一步讨论它们的基本代数性质。     

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.     

Stickelberger Codes

Let p be an odd prime and be a primitive p th root of unity over . The Galois group G of over is a cyclic group of order p-1. The integral group ring [G] contains the Stickelberger ideal S _p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S _p (q) obtained as reductions of S _p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S _p (p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B _2k , , which are divisible by p. We then develop methods to compute the generator polynomial of S _p (p). This gives rise to anew algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.     

Co-Orthogonal Codes

We define, construct and sketch possible applications of a new class of non-linear codes: co-orthogonal codes, with possible applications in cryptography and parallel processing. We also describe a fast and general method for generating (non-linear) codes with prescribed dot-products with the help of multi-linear polynomials.     

Antichain Codes

We show that almost all codes satisfy an antichain condition. This states that the minimum length of a two dimensional subcode of a code C increases if the subcode is constrained to contain a minimum weight codeword. In particular, almost no code satisfies the chain condition. In passing, we study the typical behaviour of codes with respect to generalized distances and show that almost all lie on a generalized Varshamov-Gilbert bound.     

The Correspondence Between Projective Codes and 2-weight Codes

We show how to get a 1-1 correspondence between projective linear codes and 2-weight linear codes. A generalization of the construction gives rise to several new ternary linear codes of dimension six.     

基于循环码的常重复合码的构造

研究给出了一类基于循环码的常重复合码的构造,并利用指数和计算其参数.与相关的常重复合码相比,该码具有更多的码字,且渐近性较好.     

On the Cyclicity of Goppa Codes,Parity-Check Subcodes of Goppa Codes,and Extended Goppa Codes

Classical Goppa codes are a special case of Alternant codes. First we prove that the parity-check subcodes of Goppa codes and the extended Goppa codes are both Alternant codes. Before this paper, all known cyclic Goppa codes were some particular BCH codes. Many families of Goppa codes with a cyclic extension have been found. All these cyclic codes are in fact Alternant codes associated to a cyclic Generalized Reed–Solomon code. In (1989, J. Combin. Theory Ser. A 51, 205–220) H. Stichtenoth determined all cyclic extended Goppa codes with this property. In a recent paper (T. P. Berger, 1999, in “Finite Fields: Theory, Applications and Algorithms (R. Mullin and G. Mullen, Eds.), pp. 143–154, Amer. Math. Soc., Providence), we used some semi-linear transformations on GRS codes to construct cyclic Alternant codes that are not associated to cyclic GRS codes. In this paper, we use these results to construct cyclic Goppa codes that are not BCH codes, new families of Goppa codes with a cyclic extension, and some families of non-cyclic Goppa codes with a cyclic parity-check subcode.     

Almost MDS Codes

MDS codes are codes meeting the Singleton bound. Both for theory and practice, these codes are very important and have been studied extensively. Codes near this bound, but not attaining it, have had far less attention. In this paper we study codes that almost reach the Singleton bound.     

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.     

Extending MDS Codes

A q-ary (n, k)-MDS code, linear or not, satisfies n ≤ q + k − 1. A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show that an MDS code with n = q + k − 2 can be uniquely extended to a maximum length code if and only if q is even. This result is best possible in the sense that there is, for example, a non-extendable 4-ary (5, 4)-MDS code. It may be that the proof of our result is as interesting as the result itself. We provide a simple necessary and sufficient condition for code extendability. In future work, this condition might be suitably modified to give an extendability condition for arbitrary (shorter) MDS codes.Received December 1, 2003     

中国剩余码

本文利用弱区组设计和环论中的“中国剩余定理”构造了中国剩余码。这种新型线性码是将同余类环 R/I₁∩I₂∩…∩I_n中的同余类作为信息位,将 R/J_i嵌入 R/I₁∩I₂∩…∩I_n,视其为R/I₁∩I₂∩…∩I_n的子环,R/J_i在R/I₁∩I相似文献   

Self-Orthogonal Greedy Codes

In this paper, we examine greedily generated self-orthogonal codes. Weshow that they are linear, and discuss several properties whichthey possess. We also look at specific examples of these codes.     

Codes within codes

Summary We prove coding theorems for list codes for compound channels and for codes within codes. The theorems imply corresponding results for what are usually called simply codes, which are list codes where one is the maximum decoding list length permitted to the decoder. The Bergmans coding theorem for degraded channels ([4], Theorem 15.2.1) and the positive part of the Wyner-Ziv theorem ([4], Theorem 13.2.1) are easy consequences of our results.Research supported by the National Science Foundation under Grant No. MCS 78-02148     

Almost Affine Codes

An almost affine code is a code C for which the size of all codes obtained by multiple puncturing of C is a power of the alphabet size. Essentially, almost affine codes are the same as ideal perfect secret haring schemes or partial affine geometries. The present paper explores these interrelations, gives short proofs of known and new results, and derives some properties of the distance distribution of almost affine codes.     

Binary Fingerprinting Codes

Three binary fingerprinting code classes with properties similar to codes with the identifiable parent property are proposed. In order to compare such codes a new combinatorial quality measure is introduced. In the case of two cooperating pirates the measure is derived for the proposed codes, upper and lower bounds are constructed and the results of computer searches for good codes in the sense of the quality measure are presented. Some properties of the quality measure are also derived.AMS classification:94B60, 94B65