Minimum distance and idempotent generators of minimal cyclic codes of length ${p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}$

Abstract

Let $ p_1, p_2, p_3, q $ be distinct primes and $ m={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}$. In this paper, it is shown that the explicit expressions of primitive idempotents in the semi-simple ring $R_m = { F_q[x]}/{(x^m-1)}$ are the trace function of explicit expressions of primitive idempotents from $R_{p_i^{\alpha_i}}$. The minimal polynomials, generating polynomials and minimum distances of minimal cyclic codes of length $m$ over $F_q$ are also discussed. All the results obtained in [1], [3], [4], [5], [11] and [14] are simple corollaries to the results obtained in the paper.

Received: 21 September 2020 | Accepted: 08 March 2021