Quadratic Residue Codes over F_p+vF_p and their Gray Images

TL;DR

This paper introduces quadratic residue codes over the ring Fp + vFp, explores their structure, and demonstrates their potential to generate good p-ary codes and self-dual codes, with several optimal examples.

Contribution

It extends quadratic residue code theory to rings Fp + vFp and analyzes their properties, including the construction of optimal self-dual codes and p-ary codes.

Findings

Existence of Euclidean and Hermitian self-dual extended codes for p=2

Generation of many good p-ary codes from these ring-based codes

Identification of two optimal Hermitian self-dual codes

Abstract

In this paper quadratic residue codes over the ring Fp + vFp are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes share similar properties with quadratic residue codes over finite fields. For the case p = 2, Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two optimal Hermitian self-dual codes are obtained as examples. Moreover, a substantial number of good p-ary codes are obtained as images of quadratic residue codes over Fp +vFp in the cases where p is an odd prime. These results are presented in tables.