On the Number of Cyclic Codes Over \(\mathbb{Z}_{31}\)
John Joseph O. Ondiany *
Department of Mathematics and Actuarial Sciences, Kisii University, P. O. Box 1957-40200, Kisii, Kenya.
Obogi Robert Karieko
Department of Mathematics and Actuarial Sciences, Kisii University, P. O. Box 1957-40200, Kisii, Kenya.
Lao Hussein Mude
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya.
Fred Nyamitago Monari
Department of Mathematics and Actuarial Sciences, Kisii University, P. O. Box 1957-40200, Kisii, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Let n be a positive integer, yn - 1 cyclotomic polynomial and Zq be a given finite field. In this study we determined the number of cyclic codes over \(\mathbb{Z}_{31}\). First, we partitioned the cyclotomic polynomial yn - 1 using cyclotomic cosets 31 mod n and factorized yn - 1 using case to case basis. Each monic divisor obtained is a generator polynomial and generate cyclic codes. The results obtained are useful in the field of coding theory and more especially, in error correcting codes.
Keywords: Code, cyclic codes, cyclotomic coset