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

References

• [1] S. K. Arora, M. Pruthi, Minimal cyclic codes of length 2pn, Finite Fields and Their Applications 5(2) (1999) 177–187.
• [2] G. K. Bakshi, S. Gupta, I. B. S. Passi, The algebraic structure of finite Metabelian group algebras, Communications in Algebra 43(6) (2015) 2240–2257.
• [3] G. K. Bakshi, M. Raka, Minimal cyclic codes of length $p^nq$, Finite Fields and Their Applications 9(4) (2003) 432–448.
• [4] G. K. Bakshi, M. Raka, A. Sharma, Idempotent generators of irreducible cyclic codes, In Number Theory & Discrete Geometry 6 (2008) 13–18.
• [5] S. Batra, S. K. Arora, Some cyclic codes of length 2pn, Designs Codes Cryptography 61 (2011) 41–69.
• [6] O. Broche, A. Del Río, Wedderburn decomposition of finite group algebras, Finite Fields and Their Applications 13(1) (2007) 71–79.
• [7] B. Chen, H. Liu, G. Zhang, A class of minimal cyclic codes over finite fields, Designs Codes Cryptography 74 (2013) 285–300.
• [8] R. A. Ferraz, P. M. César, Idempotents in group algebras and minimal abelian codes, Finite Fields and Their Applications 13(2) (2007) 382–393.
• [9] S. Gupta, Finite Metabelian group algebras, International Journal of Pure Mathematical Sciences 17 (2016) 30–38.
• [10] P. Kumar, S. K. Arora, $\lambda$-Mapping and primitive idempotents in semisimple ring ${\Re _{\;m}},$ Communications in Algebra 41(10) (2013) 3679-3694.
• [11] P. Kumar, S. K. Arora, S. Batra, Primitive idempotents and generator polynomials of some minimal cyclic codes of length $p^n q^m$, International Journal of Information and Coding Theory 4 (2014) 191–217.
• [12] F. J. MacWilliam, N. J. A. Sloane, The theory of error-correcting codes, edition = 1st, Elsevier, Amsterdam (2013).
• [13] M. Pruthi, S. K. Arora, Minimal codes of prime-power length, Finite Fields and Their Applications 3(2) (1977) 99–113.
• [14] A. Sahni, P. T. Sehgal, Minimal cyclic codes of length $p^nq$, Finite Fields and Their Applications 18(5) (2012) 1017–1036.
• [15] A. Sharma, G. K. Bakshi, V. C. Dumir, M. Raka, Cyclotomic numbers and primitive idempotents in the ring $GF(q)[x]/(x^{p^n}-1)$, Finite Fields and Their Applications 10(4) (2004) 653-673.