Enumeration of Cyclic Codes Over GF(23)
Kimtai Boaz Simatwo *
Department of Mathematics and Actuarial Science, Kisii University, P. O. Box 408-40200, Kisii, Kenya.
Runji Flora Mati
Department of Mathematics, Statistics and Actuarial Sciences, Karatina University, P. O. Box 1957-10101, Karatina, Kenya.
Obogi Robert Karieko
Department of Mathematics and Actuarial Science, Kisii University, P. O. Box 408-40200, Kisii, Kenya.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we investigate the number of irreducible polynomials of \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\) over GF(23). First, We factorize \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\) into irreducible polynomials over GF(23) using the cyclotomic cosets of 23 modulo \(\mathit{n}\). The number of irreducible polynomial factors of \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\) over GF(23) is equal to the number of cyclotomic cosets of 23 modulo n and each monic divisor of \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\) is a generator polynomial of cyclic codes in GF(23). Succeedingly, we confirm that the number of cyclic codes of length \(\mathit{n}\) over \(\mathit{a}\) finite field GF(23) is equivalent to the number of polynomials that divide \(\small\langle\)\(\mathit{x}\)\(\mathit{n}\) -1\(\small\rangle\).
In conclusion, we enumerate the number of cyclic codes of length n for l \(\le\) \(\mathit{n}\) < 24 and as \(\mathit{n}\) = 23\(\mathit{k}\) for l \(\leqslant\) \(\mathit{k}\) < 24
Keywords: Code, cyclic code, cyclotomic cosets