Polyadic cyclic codes over a non-chain ring F_q[u,v]/〈 f(u),g(v), uv-vu〉
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Let f(u) and g(v) be any two polynomials of degree k and ℓ respectively (k and ℓ are not both 1), which split into distinct linear factors over F_q. Let R=F_q[u,v]/〈 f(u),g(v),uv-vu〉 be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring R. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from R^n →F^kℓ n_q which preserves duality. The Gray images of polyadic codes and their extensions over the ring R lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over F_q. Some examples are also given to illustrate this.
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