On the Generalization of the Number of Cyclic Codes Over the Prime Field GF(37)

Abstract

Research has explored the characterization of cyclic codes over GF(P), where P is prime for P ≤ 23. However, no study has characterized GF(37). Additionally, no study has generalized enumeration of the number of cyclic codes of the cyclotomic polynomials u n − 1 over GF(P). In particular, the generalization of the number of cyclic codes over GF(37) for u n − 1 is also lacking in research. This study focused on the monic irreducible polynomials of u n − 1 over the finite field GF(37) with the main objective of generalizing the enumeration of the number of distinct cyclic codes. The methodology involved determining the number ofirreducible monic polynomials of the cyclotomic polynomial u n − 1 over GF(37). These polynomials were found to correspond to the number of cyclotomic cosets of 37 mod n over GF(37). The study concluded that the number of cyclic codes over GF(37) can be generalized by NGF (37) = (37y + 1)Cxm ∀x, y, m ∈ Z +. The findings provide insights into abstract algebraic concepts in coding theory that can be used to generalize number of cyclic codes over a prime field GF(P)