an:06137925
Zbl 1258.05010
Zhu, Mingzhi; Ge, Gennian
4-\({}^{\ast}\text{GDD}(6^n)\)s and related optimal quaternary constant-weight codes
EN
J. Comb. Des. 20, No. 11-12, 509-526 (2012).
00310870
2012
j
05B07 94B60
4-*GDDs; generalized Steiner systems; constant-weight codes; room squares
Summary: Constant-weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a \(K\)-GDD (where GDD is group divisible design) with the ``star'' property, denoted by \(K\)-\({}^{\ast}\text{GDD}\), was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4-\({}^{\ast}\text{GDD}(g^n)\)s. Previously, the necessary conditions for existence were shown to be sufficient for \(g=3\), and also sufficient for \(g=6\) with prime powers \(n \equiv 3,5,7 \pmod 8\) and \(n \geq 19\). We continue to investigate the existence of 4-\({}^{\ast}\text{GDD}(6^n)\)s and show that the necessary condition for the existence of a 4-\({}^{\ast}\text{GDD}(6^n)\), namely, \(n \geq 14\), is also sufficient. The known results on the existence of optimal quaternary \((n,5,4)\) CWCs are also extended.