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4-\({}^{\ast}\text{GDD}(6^n)\)s and related optimal quaternary constant-weight codes. (English) Zbl 1258.05010
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.

MSC:
05B07 Triple systems
94B60 Other types of codes
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