Perfect single error-correcting codes in the Johnson Scheme

Delsarte conjectured in 1973 that there are no nontrivial pefect codes in the Johnson scheme. Etzion and Schwartz recently showed that perfect codes must be k-regular for large k, and used this to show that there are no perfect codes correcting single errors in J(n,w) for n <= 50000. In this paper we show that there are no perfect single error-correcting codes for n <= 2^250.Comment: 4 pages, revised, accepted for publication in IEEE Transactions on Information Theor

Some Bounds for the Number of Blocks III

Let $\mathcal D=(\Omega, \mathcal B)$ be a pair of $v$ point set $\Omega$ and a set $\mathcal B$ consists of $k$ point subsets of $\Omega$ which are called blocks. Let $d$ be the maximal cardinality of the intersections between the distinct two blocks in $\mathcal B$. The triple $(v,k,d)$ is called the parameter of $\mathcal B$. Let $b$ be the number of the blocks in $\mathcal B$. It is shown that inequality ${v\choose d+2i-1}\geq b\{{k\choose d+2i-1} +{k\choose d+2i-2}{v-k\choose 1}+....$ $.+{k\choose d+i}{v-k\choose i-1} \}$ holds for each $i$ satisfying $1\leq i\leq k-d$, in the paper: Some Bounds for the Number of Blocks, Europ. J. Combinatorics 22 (2001), 91--94, by R. Noda. If $b$ achieves the upper bound, $\mathcal D$ is called a $\beta(i)$ design. In the paper, an upper bound and a lower bound, $\frac{(d+2i)(k-d)}{i}\leq v \leq \frac{(d+2(i-1))(k-d)}{i-1}$, for $v$ of a $\beta(i)$ design $\mathcal D$ are given. In the present paper we consider the cases when $v$ does not achieve the upper bound or lower bound given above, and get new more strict bounds for $v$ respectively. We apply this bound to the problem of the perfect $e$-codes in the Johnson scheme, and improve the bound given by Roos in 1983.Comment: Stylistic corrections are made. References are adde

New Constant-Weight Codes from Propagation Rules

This paper proposes some simple propagation rules which give rise to new binary constant-weight codes.Comment: 4 page

Cocyclic simplex codes of type alpha over Z4 and Z2s

Over the past decade, cocycles have been used to construct Hadamard and generalized Hadamard matrices. This, in turn, has led to the construction of codes-self-dual and others. Here we explore these ideas further to construct cocyclic complex and Butson-Hadamard matrices, and subsequently we use the matrices to construct simplex codes of type /spl alpha/ over Z(4) and Z(2/sup s/), respectively

On the binary linear constant weight codes and their autormorphism groups

We give a characterization for the binary linear constant weight codes by using the symmetric difference of the supports of the codewords. This characterization gives a correspondence between the set of binary linear constant weight codes and the set of partitions for the union of supports of the codewords. By using this correspondence, we present a formula for the order of the automorphism group of a binary linear constant weight code in terms of its parameters. This formula is a key step to determine more algebraic structures on constant weight codes with given parameters. Bonisoli [Bonisoli, A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combinatoria 18, 181--186 (1984)] proves that the $q$-ary linear constant weight codes with the same parameters are equivalent (for the binary case permutation equivalent). We also give an alternative proof for Bonisoli's theorem by presenting an explicit permutation on symmetric difference of the supports of the codewords which gives the permutation equivalence between the binary linear constant weight codes.Comment: 12 page