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

Amita  Sahni; Poonam Trama  Sehgal

doi:10.3934/amc.2015.9.437

Article Contents

2015, Volume 9, Issue 4: 437-447. Doi: 10.3934/amc.2015.9.437

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

Amita Sahni^1, and
Poonam Trama Sehgal^1,

1.
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160014

Received: September 30, 2013

Revised: November 30, 2014

Published: October 2015

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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.

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Mathematics Subject Classification: 11T71, 94B05.

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