Roulette games and depths of words over finite commutative rings

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 R, where alpha is a non-zero element of the Teichmuller set of R, gamma is a generator of the maximal ideal of R and beta is a unit in R. As a consequence, we explicitly determine depth distributions of all repeated-root (alpha + gamma beta)-constacyclic codes of prime power lengths over R.