Semi-Distance Codes and Steiner Systems

Abstract

Let C be a d-semi-distance code of length n and N the cardinality of C. In this paper we obtain an upper bound on N: $N \le { n \choose k_0-d+1 } / { k_0 \choose k_0-d+1 }$ , where k₀  =  ⌊ (n + d − 1)/2 ⌋. When a code C attains the upper bound and n + d − 1 is even, C corresponds to a Steiner system S(k₀ − d + 1, k₀, n) in a natural way. Let S be a Steiner system S(t,k,n) with k + t − 1 ≤ n ≤ k + t + 1 (1 ≤ t ≤ k  <  n). Then S corresponds to an optimal (k − t + 1)-semi-distance code of length n in a natural way.