Let p be an odd prime, q = p m , R = F q + uF q with u 2 = 1, and S = F q + uF q + vF q + uvF q with u 2 = 1, v 2 = 1, uv = vu. In this paper, F q RS-cyclic codes over F q RS are studied. As an application, we present a construction of quantum error-correcting codes (QECCs) from the F q RS-cyclic codes over F q RS, which provides new QECCs. We also consider linear complementary dual (LCD) codes from the F q RS-cyclic codes over F q RS. Among others, we construct a Gray map over F q RS and discuss the Gray images of F q RS-cyclic codes over F q .INDEX TERMS Cyclic codes, mixed alphabet codes, QECCs, LCD codes.

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

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