Semi-distance codes and Steiner systems (Q2373934)

The \textit{semi-distance} between two binary vectors \(x\) and \(y\) of length \(n\) is the number of positions in which \(x\) is one and \(y\) is zero. A binary code \(C\) of length \(n\) is a \textit{\(d\)-semi-distance code} if \(C\) has minimum semi-distance \(d\). The authors determine an upper bound on the size of a binary \(d\)-semi-distance code \(C\) of length \(n\). When a code \(C\) attains the upper bound and \(n+d-1\) is even, then \(C\) corresponds to a Steiner system \(S(k_0-d+1,k_0,n)\), where \(k_0=(n+d-1)/2\). Vice versa, let \(S\) be a Steiner system \(S(t,k,n)\) with \(k+t-1 \leq n\leq k+t+1\) (\(1\leq t \leq k<n\)), then \(S\) corresponds to an optimal \((k-t+1)\)-semi-distance code of length \(n\). The presented results improve results of Naemura, Nakamura, and Ikeno for constant weight \(d\)-semi-distance codes.