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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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