Another q-Polynomial Approach to Cyclic Codes

c0 + c1x+ c2x 2 + · · ·+ cn−1x n−1 ∈ GF(q)[x]/(x − 1), any code C of length n over GF(q) corresponds to a subset of the quotient ring GF(q)[x]/(xn− 1). A linear code C is cyclic if and only if the corresponding subset in GF(q)[x]/(xn − 1) is an ideal of the ring GF(q)[x]/(xn − 1). It is well known that every ideal of GF(q)[x]/(xn−1) is principal. Let C = 〈g(x)〉 be a cyclic code, where g(x) is monic and has the smallest degree among all the generators of the ideal C . Then g(x) is unique and called the generator polynomial, and h(x) = (xn − 1)/g(x) is referred to as the parity-check polynomial of C . Cyclic codes are widely employed in consumer electronics, data transmission devices, broadcast systems, and computer systems as they have efficient encoding and decoding algorithms. Cyclic codes have been studied for decades and a lot of progress has been made (see, for example, [1, 3, 7, 8, 4, 6] and the references therein). Three approaches are generally used in the design and analysis of cyclic codes, and based on a generator matrix, a generator polynomial and a generating idempotent, respectively. These approaches have their advantages and disadvantages in dealing with cyclic codes. Recently, a q-polynomial approach to the construction and analysis of cyclic codes over GF(q) was given by Ding and Ling [2]. Further progress was made in [5]. The objective of this paper is to present another q-polynomial approach to all cyclic codes over GF(q).