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IEEE Transactions on Information Theory

Quantum codes from cyclic codes over GF(4/sup m/)

Andrew Thangaraj, S.W. McLaughlin

Published Mar 1, 2001• 45 citations
Publisher

Abstract

We provide a construction for quantum codes (Hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4/sup m/). We also provide examples of these codes some of which meet the known bounds for quantum codes.
DOI
10.1109/18.915675

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The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.
In this paper, we introduce a class of dualcontaining 2 -generator quasi-cyclic codes over GF(4)G F(4)GF(4). By Hermitian construction, we show computation constructions of binary quantum codes with good parameters from quaternary quasi-cyclic codes. Furthermore, 3 new quantum codes [[42, 10, 9]], [[74,,36, 9]] and [[78, 40, 9]] are derived from Hermitian dual-containing quasi-cyclic codes by an exhaustive search, along with 24 newly derived codes that enhance the previously best-known lower bounds on minimum distance.
QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER F4+uF4\mathbb{F}_4 + u\mathbb{F}_4F4​+uF4​