The number of self-dual cyclic codes over finite fields

In linear coding theory, self-dual cyclic codes are especially notable for their efficiency in both encoding and decoding processes. This research focuses on the enumeration of such codes over finite fields, denoted as \({\mathbb {F}}_q\), where \(q = 2^m\) and m is the field size. Traditionally, investigations in this area have faced significant constraints primarily due to two factors. The first is the length of the code, n, with a focus on excluding prime factors congruent to 1 modulo 8. The second limitation pertains to the binary case, specifically when \(m = 1\). To overcome these challenges, this study introduces the concept of the exact power character of 2, a novel approach that offers a significant methodological advancement. By reframing the existing numerical constraints in terms of three readily computable parameters, this approach effectively sidesteps the limitations previously existing in the field. This development not only broadens the scope of possible code lengths and field sizes but also enhances the potential for practical applications of self-dual cyclic codes in various areas of information theory and communications.