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

Year 2021, Volume: 8 Issue: 3, 167 - 195, 15.09.2021

Pankaj Kumar Pinki Devi

https://doi.org/10.13069/jacodesmath.1000837

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 \cite{ref[1]}, \cite{ref[4]}, \cite{ref[5]}, \cite{ref[6]}, \cite{ref[11]} and \cite{ref[14]} are simple corollaries to the results obtained in the paper.

Keywords

Cyclotomic cosets, Primitive idempotents, Cyclic codes, Trace function

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