Cyclic codes over \(R_{k}\)
Afer a preliminary introduction, this paper is organized as follows: In Section 2, the authors give necessary background on linear and cyclic codes over the rings \(R_k\). In Section 3, the authors describe cyclic codes over \(R_k\) using the decompositions of the polynomial \(x^n-1\) over the ring \(R_k\). The ideals of the ring \(R_k[x]/(x^n-1)\) are described and a formula for the number of cyclic codes over \(R_k\) of certain lengths is given. Also, necessary and sufficient conditions on \(n\) are given for \(R_k[x]/(x^n-1)\) to be a local ring. In Section 4, one-generator cyclic codes over the ring \(R_k\) are classified. In particular, the authors obtain necessary and sufficient conditions on the coefficients of a polynomial in \(R_k[x]/(x^n-1)\) to generate a non-trivial cyclic code. These ideas are later used to construct some specific one-generator cyclic codes over \(R_k\) whose binary images are optimal or near-optimal. The images of cyclic codes over \(R_k\) are described in Section 5. At first, the authors consider the \(R_{k-1}\) images of odd-length cyclic codes over \(R_k\), and they prove that these are equivalent under Nechaev permutation to cyclic codes. Later, the authors prove that the binary images of cyclic codes over \(R_k\) under the Gray map \(\psi_k\) are binary quasi-cyclic codes of degree \(2^k\). The paper is concluded with some remarks and directions for future work.
scientific article; zbMATH DE number 6010508
scientific article