Explicit factorization of xl1m1l2m2l3m3−a and a-constacyclic codes over a finite field

Abstract

Let ${\mathbb{F}}_q$ be a finite field of order q, t be a prime and $m_1,m_2,m_3$ be positive integers. In this article, we find all irreducible divisors of $x^{l_1^{m_1}{l}_2^{m_2}{l}_3^{m_3}}-a$ over ${\mathbb{F}}_q$ where $a\in {\mathbb{F}}_q^{*}$ and $q^t-1=l_1^{v_1}{l}_2^{v_2}{l}_3^{v_3}c$ such that $l_1,l_2,l_3$ are distinct odd primes and c is a positive integer with $\gcd (l_1{l}_2{l}_3,c)=1$ and $\gcd (l_1{l}_2{l}_3,q(q-1))=1.$ Moreover, we construct an a-constacyclic code by using these irreducible divisors.

Keywords:

• Constacyclic codes
• cyclotomic coset
• generator polynomials
• irreducible factor polynomials

2020 Mathematics Subject Classification:

• 94B05
• 94B15

Acknowledgments

The authors would like to thank the referees for their useful suggestions.

Additional information

Funding

The first author was supported by Science Achievement Scholarship of Thailand (SAST). The fourth author was supported by National Research Council of Thailand (Grant No. NRCT5-RSA63011-05).