Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields
Amita Sahni, Poonam Trama Sehgal · Advances in Mathematics of Communications · 2015
The main objective of this article is to study self-orthogonal negacycliccodes of length $n$ over a finite field $\mathbb{F}_{q}$, wherethe characteristic of $\mathbb{F}_{q}$ does not divide $n$. We investigateissues related to their existence, characterization and enumeration.We find the necessary and sufficient conditions for the existenceof self-orthogonal negacyclic codes of length $n$ over a finite field$\mathbb{F}_{q}$. We characterize the defining sets and the correspondinggenerator polynomials of these codes. We obtain formulae to calculatethe number of self-dual and self-orthogonal negacyclic codes of agiven length $n$ over $\mathbb{F}_{q}$. The enumeration formulafor 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 necessaryand sufficient conditions when $\chi(d,q)=0$ holds.