On the Generalization of the Number of Cyclic Codes Over the Prime Field GF(37)
Pancras Ongili *
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Lao Hussein Mude
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Kinyanjui Jeremiah Ndung’u
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Research has explored the characterization of cyclic codes over GF(P), where P is prime for P \(\le\) 23. However, no study has characterized GF(37). Additionally, no study has generalized enumeration of the number of cyclic codes of the cyclotomic polynomials un - 1 over GF(P). In particular, the generalization of the number of cyclic codes over GF(37) for un - 1 is also lacking in research. This study focused on the monic irreducible polynomials of un - 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 of irreducible monic polynomials of the cyclotomic polynomial un - 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 \(\forall\)x, y, m \(\in\) 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).
Keywords: Generalization over GF(37), GF(P), u\(^n\)-1, irreducible factors, cyclotomic cosets, cyclotomic polynomials, cyclic codes