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arXiv:alg-geom/9711026

Symmetric matrices and quantum codes

Vladimir D. Tonchev

Published Nov 20, 1997• 0 citations
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Abstract

This paper has been withdrawn since a Gilbert-Varshamov bound for general quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett. 77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with orthogonal geometry, or equivalently, with symmetric matrices as in this paper, has been proved by Calredbank, Rains, Shor and Sloane, Phys. Rev. Lett. 78, p. 405. I would like to thank Robert Calderbank for pointing out these references to me.

DOI
10.48550/arxiv.alg-geom/9711026

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We construct asymptotically good nested Calderbank-Shor-Steane (CSS) code pairs from Hsu-Anastasopoulos codes and MacKay-Neal codes. For fixed balanced triples with even component degrees and jZ≥4j_Z\ge4jZ​≥4, we prove that the coding rate stays bounded away from zero and that the relative distances on both sides stay bounded away from zero with probability tending to one as the blocklength grows. Moreover, within an explicit low-degree search window, we prove that all 56 balanced triples in that window with even common row degree, column degree at least four, and positive design quantum rate attain the classical Gilbert-Varshamov (GV) bound on both constituent sides, and consequently the CSS GV bound at fixed finite degree.