Further factorization of xⁿ-1 over finite fields (II)

2020/12/15 by Yansheng Wu, Qin Yue, Wu, Yansheng +1
Computer Science · Engineering · Social Sciences · #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #Islamic Finance and Communication #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2012.08161

openalex publication_date 2020/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \Bbb F_q be a finite field with q elements. Let n be a positive integer with radical rad(n), namely, the product of distinct prime divisors of n. If the order of q modulo rad(n) is either 1 or a prime, then the irreducible factorization and a counting formula of irreducible factors of xⁿ-1 over \Bbb F_q were obtained by Mart'ınez, Vergara, and Oliveira (Des Codes Cryptogr 77 (1) : 277-286, 2015) and Wu, Yue, and Fan (Finite Fields Appl 54: 197-215, 2018). In this paper, we explicitly factorize xⁿ-1 into irreducible factors in \Bbb F_q[x] and calculate the number of the irreducible factors when the order of q modulo rad(n) is a product of two primes.

Citations

• A construction of linear codes and their complete weight enumerators

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