Isometrically Self-dual Cyclic Codes

TL;DR

This paper introduces the concept of isometrically self-dual cyclic codes, providing conditions for their existence, a construction method, and examples including a class of MDS cyclic codes derived from Reed-Solomon codes.

Contribution

It defines isometrically self-dual cyclic codes, establishes existence conditions via Type-I duadic splittings, and offers a construction method with concrete examples.

Findings

Necessary and sufficient conditions for existence.

Construction method for isometrically self-dual cyclic codes.

Examples including a class of MDS cyclic codes.

Abstract

General isometries of cyclic codes, including multipliers and translations, are introduced; and isometrically self-dual cyclic codes are defined. In terms of Type-I duadic splittings given by multipliers and translations, a necessary and sufficient condition for the existence of isometrically self-dual cyclic codes is obtained. A program to construct isometrically self-dual cyclic codes is provided, and illustrated by several examples. In particular, a class of isometrically self-dual MDS cyclic codes, which are alternant codes from a class of generalized Reed-Solomon codes, is presented.