On Counting the Number of Cyclic Codes of Length N over Prime Fields

Abstract

Cyclic codes are a cornerstone of coding theory. Many studies have extensively explored specific finite prime fields, GF(p). However, a generalized framework for GF(p) remains unexplored. Previous research has enumerated cyclic codes over GF(13), GF(17), GF(19), GF(23), GF(31), and GF(37), deriving significant find- ings for special cases where n assumes specific forms like n = pm or n = am · pm . Despite these advances, existing studies are often restricted to individual prime fields and lack a unified framework applicable across all GF(p). This research de- velops a mathematical framework for counting cyclic codes over any prime field GF(p). The methodology involves use of rigorous mathematical derivations and proofs to establish the relationship between number of cyclic codes, cyclotomic cosets and the factorization of xn − 1 into irreducible polynomials over GF(p). This framework provides a comprehensive understanding of how the structures of n and p influence the number of cyclic codes. Results show that the number of cyclic codes, N, over the prime fields can be expressed as N = (py + 1)C , where C is derived from the number of cyclotomic cosets modulo k ∀ n = k · py . This work contributes to efficient design of error-correcting codes, strengthens the theoretical foundation for secure communication systems, and bridges theoretical concepts of pure mathematics.