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article number 1750161International audienceGiven a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. The first type, pseudo-self-dual bases, are a generalization of trace orthogonal bases for fields. The second are called symmetric bases. Both types are illustrated with skew cyclic codes which are codes that are $A$-submodules of the skew polynomial ring $A[X;\theta]/\langle X^n-1\rangle$ (the classical cyclic codes are the case when $\theta=id$). When $A$ is commutative, there exists criteria for a skew cyclic code over $A$ to be self-dual. With this criteria and a duality preserving map, many self-dual codes over the subring $R$ can easily be found. In this fashion, numerous examples are given, some of which are not chain or serial rings

Duality preserving Gray maps for codes over rings