On Counting the Number of Cyclic Codes of Length n Over Prime Fields

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 findings for special cases where n assumes specific forms like $n = p^m$ or $n = a^m \cdot p^m$. Despite these advances, existing studies are often restricted to individual prime fields and lack a unified. framework applicable across all GF(p). This research develops a mathematical framework for counting cyclic codes over any prime field GF(p). The methodology involves the use of rigorous mathematical derivations and proofs to establish the relationship between number of cyclic codes, cyclotomic cosets and the factorization of $x^n - 1$ 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. The study concludes that the number of cyclic codes, $\mathcal{N}$, over the prime fields can be expressed as $\mathcal{N} = (p^y + 1)^C$, where C is derived from the number of cyclotomic cosets modulo $k \forall n = k \cdot p^y $. This work contributes to the efficient design of error-correcting codes, strengthens the theoretical foundation for secure communication systems, and bridges theoretical concepts of pure mathematics.