Some results on cyclic codes over the ring \(R_{2,m}\) (Q2873183)

Denoting the ring \(\mathbb{F}_{2^m}[u_1, u_2,\ldots,u_k]/<u_i^2, u_iu_j-u_ju_i|i,j=1,\ldots,k>\) as \(R_{k,m}\) (and \(\mathbb{F}_{2^m}\) as \(R_{0,m}\)), the authors introduce a unique set of generators for cyclic codes of arbitrary length over \(R_{2,m}\) (continuing preliminary work of [\textit{B. Yildiz} and \textit{S. Karadeniz}, Des. Codes Cryptography 58, No. 3, 221--234 (2011; Zbl 1213.94182)], thus giving a complete classification of such codes. An algorithm generating all cyclic codes of given length over \(R_{2,m}\) is presented; especially, all distinct cyclic codes of length 2 over \(R_{2,m}\) , i.e. ideals of \(R_{2,m}[x]/<x^2-1>\), are found and their number is given.NEWLINENEWLINE By defining a one-to-one correspondence between cyclic codes of length \(2n\), \(n\) odd, over \(R_{k-1,m}\) and cyclic codes of length \(n\) over \(R_{k,m}\), the authors determine the number of ideals over the rings \(R_{2,m}\) and \(R_{3,m}\) (thus solving parts of a research problem posed by \textit{S. T. Dougherty} et al. [Des. Codes Cryptography 63, No. 1, 113--126 (2012; Zbl 1237.94134)] and, as a corollary, the number of cyclic codes of odd length \(n\) over \(R_{2,m}\) and \(R_{3,m}\).