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arXiv:cs/0701060 · IEEE International Symposium on Information Theory (ISIT)

Duadic Group Algebra Codes

Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli

Published Jun 1, 2007• 13 citations
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Abstract

Duadic group algebra codes are a generalization of quadratic residue codes. This paper addresses an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction.

DOI
10.1109/isit.2007.4557530

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Duadic group algebra codes are a generalization of quadratic residue codes. This paper addresses an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum stabilizer codes that have the nice feature that many errors of small weight do not need error correction.
Good quantum codes, such as quantum MDS codes, are typically nondegenerate, meaning that errors of small weight require active error-correction, which is -unfortunately- itself prone to errors. In this paper, examples of degenerate quantum codes are constructed that alleviate this problem in that they allow some errors of small weight that do not require active error correction. In particular, two new families of [[n, 1, ges radicn]]qdegenerate quantum codes are derived from classical duadic codes
We present a universal framework for quantum error-correcting codes, i.e., a framework that applies to the most general quantum error-correcting codes. This framework is based on the group algebra, an algebraic notation associated with nice error bases of quantum systems. The nicest thing about this framework is that we can characterize the properties of quantum codes by the properties of the group algebra. We show how it characterizes the properties of quantum codes as well as generates some new results about quantum codes.
In this paper, we illustrate how to simplify the constructions of quantum error-correction codes via the quadratic residues. The suggested quantum code, which is the stabilizer quantum code generated from an Abelian group with the basis being commutative quantum operators selected from rows of an Pauli block matrix, does not require the dual-containing (or self-orthogonal) constraint necessary for the standard quantum error-correction code, thus allowing us to construct a quantum code with the large codeword length.