Enumeration formulas for self-dual cyclic codes

Self-conjugate-reciprocal irreducible monic factors of x − 1 over finite fields and their applications

“…In the study of SRIM and SCRIM factors of x n ±1, it is therefore sufficient to assume that gcd(n, q) = 1. Over a finite field of even characteristic, x n +1 = x n −1 and its SRIM and SCRIM factors were completely studied in [2] and [4]. Without loss of generality, q is assume to be an odd prime power throughout the study SRIM and SCRIM factors of x n + 1 over F q and F q 2 , respectively.…”

“…In this subsection, we focus on recursive enumeration for SRIM factors of x n + 1 over F q . It can be given in terms of the number of SRIM factors of x n ′ − 1 which is determined [4], where n ′ is the largest odd divisor of n. A recursive formula for the number of SRIM factors of x n + 1 over F q is given as follows.…”

On self-duality and hulls of cyclic codes over $$\frac{\mathbb {F}_{2^m}[u]}{\langle u^k\rangle }$$ with oddly even length

“…By Corollary 4.1, we can get a class of self-dual and 2-quasi-cyclic codes over F 2 m of length 4n from the class of self-dual cyclic code over R of length 2n and the Gray map φ defined by Equation (8). In the following, we consider how to give an efficient encoder for each self-dual and 2-quasi-cyclic code φ(C) of length 4n over F 2 m derived from a self-dual cyclic code C of length 2n over…”

“…Finally, by Lemma 4.2 and Theorem 4.3 we obtain 945 binary self-dual and 2-quasi-cyclic codes φ(C) of length 60. For example, among these codes we have the following 48 self-dual and 2-quasi-cyclic codes φ(C) with basic parameters [60,30,8], which are determined by:…”

“…In 2016, Chen et al [8] given some new necessary and sufficient conditions for the existence of nontrivial self-dual simple-root cyclic codes over finite commutative chain rings and studied explicit enumeration formulas for these codes. But self-dual repeated-root cyclic codes over finite commutative chain rings were not considered in [8].…”