Constacyclic codes over Fq[u1, u2, ..., uk]/(u3i - ui, uiuj - ujui ) and their applications of constructing quantum codes

In this paper, we study constacyclic codes over the ring R-k = F-q[u(1), u(2),middot middot middot , u(k)]/(u(i)(3)- u(i), u(i)u(j) - u(j)u(i)), for all i, j = 1, . . . , k, q = p(m) for any odd prime p and positive integers m, k. We study the structure of the ring R-k and define a Gray map by a matrix and decompose constacyclic codes over the ring R-k as a direct sum of constacyclic codes over F-q. We give the relevant properties of constacyclic codes over the ring R-k and give necessary and sufficient conditions for constacyclic codes to be dual-containing. Taking R-2 as an example, we give some corresponding conclusions. As an application, we construct some new quantum error-correcting codes over F-q from constacyclic codes over R-k.