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

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

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