Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields

Abstract

The main objective of this article is to study self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$, where the characteristic of $\mathbb{F}_{q}$ does not divide $n$. We investigate issues related to their existence, characterization and enumeration. We find the necessary and sufficient conditions for the existence of self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$. We characterize the defining sets and the corresponding generator polynomials of these codes. We obtain formulae to calculate the number of self-dual and self-orthogonal negacyclic codes of a given length $n$ over $\mathbb{F}_{q}$. The enumeration formula for self-orthogonal negacyclic codes involves a two-variable function $\chi(d,q)$ defined by $\chi(d,q)=0$ if $d$ divides $(q^{k}+1)$ for some $k\geq0$ and $\chi(d,q)=1$, otherwise. We give necessary and sufficient conditions when $\chi(d,q)=0$ holds.