On the complete weight enumerators of some linear codes with a few weights

Linear codes with a few weights have important applications in authentication codes, secret sharing, consumer electronics, etc.. The determination of the parameters such as Hamming weight distributions and complete weight enumerators of linear codes are important research topics. In this paper, we consider some classes of linear codes with a few weights and determine the complete weight enumerators from which the corresponding Hamming weight distributions are derived with help of some sums involving Legendre symbol

Near MDS poset codes and distributions

We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the "ordered metric" characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.Comment: 13 pages, 1 figur

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable. Keywords: 1-perfect code; triply-shortened 1-perfect code; equitable partition; perfect coloring; weight distribution; distance distributionComment: 12 page

Bounds on the size of codes

In this dissertation we determine new bounds and properties of codes in three different finite metric spaces, namely the ordered Hamming space, the binary Hamming space, and the Johnson space. The ordered Hamming space is a generalization of the Hamming space that arises in several different problems of coding theory and numerical integration. Structural properties of this space are well described in the framework of Delsarte's theory of association schemes. Relying on this theory, we perform a detailed study of polynomials related to the ordered Hamming space and derive new asymptotic bounds on the size of codes in this space which improve upon the estimates known earlier. A related project concerns linear codes in the ordered Hamming space. We define and analyze a class of near-optimal codes, called near-Maximum Distance Separable codes. We determine the weight distribution and provide constructions of such codes. Codes in the ordered Hamming space are dual to a certain type of point distributions in the unit cube used in numerical integration. We show that near-Maximum Distance Separable codes are equivalently represented as certain near-optimal point distributions. In the third part of our study we derive a new upper bound on the size of a family of subsets of a finite set with restricted pairwise intersections, which improves upon the well-known Frankl-Wilson upper bound. The new bound is obtained by analyzing a refinement of the association scheme of the Hamming space (the Terwilliger algebra) and intertwining functions of the symmetric group. Finally, in the fourth set of problems we determine new estimates on the size of codes in the Johnson space. We also suggest a new approach to the derivation of the well-known Johnson bound for codes in this space. Our estimates are often valid in the region where the Johnson bound is vacuous. We show that these methods are also applicable to the case of multiple packings in the Hamming space (list-decodable codes). In this context we recover the best known estimate on the size of list-decodable codes in a new way