Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉

In this paper, we study linear codes over ring Rk = 𝔽pm[u1, u2,⋯,uk]/〈$\begin{array}{} u^{2}_{i} \end{array} $ = ui, uiuj = ujui〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from $\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $ and give the generator polynomials of constacyclic codes over Rk. We also study the MacWilliams identities of linear codes over Rk.

On the linear codes over the ring Rp

Some results are generalized on linear codes over [Formula: see text] in [15] to the ring [Formula: see text], where [Formula: see text] is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over [Formula: see text] are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over [Formula: see text]. A nontrivial automorphism [Formula: see text] on the ring [Formula: see text] is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over [Formula: see text] are introduced. The number of distinct skew cyclic codes over [Formula: see text] is given. The Gray images of skew codes over [Formula: see text] are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over [Formula: see text] are introduced. MacWilliams identities of linear codes over [Formula: see text] are given.

THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES

We use derivatives to prove the equivalences between MacWilliams identity and its four equivalent forms, and present new interpretations for the four equivalent forms.