Roulette games and depths of words over finite commutative rings

Abstract

In this paper, we propose three new turn-based two player roulette games and provide positional winning strategies for these games in terms of depths of words over finite commutative rings with unity. We further discuss the feasibility of these winning strategies by studying depths of codewords of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over a finite commutative chain ring $${\mathcal {R}},$$ where $$\alpha $$ is a non-zero element of the Teichmuller set of $${\mathcal {R}},$$ $$\gamma $$ is a generator of the maximal ideal of $${\mathcal {R}}$$ and $$\beta $$ is a unit in $${\mathcal {R}}.$$ As a consequence, we explicitly determine depth distributions of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over $${\mathcal {R}}$$ .