This paper investigates general properties of codes with the rank metric. First, we investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their parameters, and investigate their asymptotic covering properties. Finally, we establish several identities that relate the rank weight distribution of a linear co...

In this article, a differential construction for the dual of an algebraic-geometric code on a projective surface is given. Afterwards, this result is used to lower bound the minimum distance of this dual code. The found bounds involve intersection numbers of some particular divisors on the surface. AMS Classification: 14J20, 94B27, 11G25.

Let Fpm be a finite field with cardinality p and R = Fpm + uFpm with u = 0. We aim to determine all α + uβ-constacyclic codes of length np over R, where α, β ∈ Fpm , n, s ∈ N+ and gcd(n, p) = 1. Let α0 ∈ F ∗ pm and α s 0 = α. The residue ring R[x]/〈x np − α − uβ〉 is a chain ring with the maximal ideal 〈x − α0〉 in the case that x n − α0 is irreducible in Fpm[x]. If x−α0 is reducible in Fpm[x], w...

As far as we know, there is no decoding algorithm of any binary self-dual [40, 20, 8] code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual [40, 20, 8] code is not efficient in the sense that it cannot be done by hand due to a large syndrome table. The purpose of this paper is to give two new efficient decoding algorithms for an extrem...

We show that a construction described in Clark, Key and de Resmini [9] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p where p is a prime, and m ≥ 1. This gives words of weight 2p + 1 − p −1 p−1 in the dual of the p-ary code of the desarguesian plane of order p, and provides an im...

where H i m ) = -mlog2 ( U ) (1 crjlog, (1 a). In this correspondence we present new estimates of the relative error term improving the known results. Our approach is based on investigating upper bounds for absolute values of Krawtchouk polynomials. Actually, this is just a refinement of an idea due to P. Sol6 [ 161. The bounds obtained are quite tight. Particularly, for t fixed our estimate is...

A method is given for the construction of linear codes with prescribed minimum distance and also prescribed minimum distance of the dual code. This works for codes over arbitrary finite fields. In the case of binary codes Matsumoto et al. showed how such codes can be used to construct cryptographic Boolean functions. This new method allows to compute new bounds on the size of such codes, extend...

Gleason has described the general form that the weight distribution of a self-dual code over GF(2) and GF(3) can have. We give an explicit formula for this weight distribution when the minimum distance d between codewords is made as large as possible. It follows that for self-dual codes of length n over GF(2) with all weights divisible by 4, d ~ 4[n/24] + 4; and for self-dual codes over GF(3), ...

We study s-extremal codes over F4 or over F2. A Type I self-dual code over F4 or over F2 of length n and minimum distance d is s-extremal if the minimum weight of its shadow is largest possible. The purpose of this paper is to give some results which are missing in a series of papers by Bachoc and Gaborit [2], by Gaborit [6], and by Bautista, et. al. [1]. In particular, we give an explicit form...

An [18, 9, 8] extremal self-dual code over GF( 4 ) and related [17, 4, 12], [17, 8, 8],[17, 9, 7], [17, 13, 4] codes are investigated. Simple coordinates are given for these codes,and their minimal weight codewords and automorphism groups are determined. Thegroups are all isomorphic to 3 × (PSL 2 ( 16 ). 4 ). __________* This appeared in ‘‘Discrete Math.’’, vol. 83 (1990), pp. 2...