Constant weight codes and group divisible designs
This paper continues the study of optimal constant weight codes over arbitrary alphabets which was initiated by \textit{T. Etzion} [Optimal constant weight codes over \(Z_k\) and generalized designs, Discrete Math. 169, No. 1-3, 55-82 (1997)]. Etzion showed that such codes are equivalent to special GDD's known as generalized Steiner systems \(\text{GS}(t, k,n,g)\). This paper presents recursive constructions for these designs when \(t=2\) and \(k=3\). These constructions allow the authors to prove that for a fixed group size \(g\), the spectrum of \(\text{GS}(2,3,n,g)\) is PBD-closed. This enables them to show that the necessary numerical conditions on the values of \(n\) are asymptotically sufficient. Since this paper was published subsequent work has concentrated on finding the exact spectrum of \(n\) for which there exists a \(\text{GS}(2,3,n,g)\) for a fixed small group size \(g\).
scientific article; zbMATH DE number 1275100