Quantum MDS and Synchronizable Codes From Cyclic and Negacyclic Codes of Length 2ps  Over F pm

Hai Q Dinh,Woraphon Yamaka,Bac T Nguyen

doi:10.1109/access.2020.3006001

Abstract

Let p be an odd prime, and F p(m) is the finite field of p m elements. In this paper, all maximum distance separable (briefly, MDS) cyclic and negacyclic codes of length 2p s over F p(m) are established. As an application, all quantum MDS (briefly, qMDS) codes are constructed from cyclic and negacyclic codes of length 2p s over finite fields using the Calderbank Shor-Steane (briefly, CSS) and Hermitian constructions. These codes are new in the sense that their parameters are different from all the previous constructions. Furthermore, quantum synchronizable codes (briefly, QSCs) are obtained from cyclic codes of length 2p s over F p(m) . To enrich the variety of available QSCs, many new QSCs are constructed to illustrate our results. Among them, there are QSCs codes with shorter lengths and much larger minimum distances than known primitive narrow-sense Bose-Chaudhuri-Hocquenghem (briefly, BCH) codes.

Highlights

Let p be a prime number and Fpm a finite field
Applying Theorem 17, all qMDS codes constructed from Ci,j using the CSS construction are determined in the following table
We give all qMDS codes constructed from Ci,j using the Hermitian construction

Summary

INTRODUCTION

Let p be a prime number and Fpm a finite field. An [n, k] linear code C over Fpm is a k-dimensional subspace of Fnpm. Transformed the problem of finding QEC codes from classical error-correcting codes over GF(4) [9]. Calderbank et al introduced a method to construct QEC codes from classical error-correcting codes [9]. Entanglement-assisted quantum error-correcting (briefly, EAQEC) codes are considered as a new research direction of quantum coding theory. Motivated by [5] and [77], Luo and Ma [59] proposed a general construction of QSCs with CSS structure from classical dual-containing cyclic codes and obtained a distance bound using a rational function for the proposed QSCs. Quantum noise is described by operators that act on qubits, with the most general model being the linear combinations of the Pauli operators I , X , Y , and Z acting on each qubit individually.

PRELIMINARIES

QUANTUM MDS CODES

QUANTUM SYNCHRONIZABLE CODES

CONCLUSION