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International Journal of Quantum Information

QUANTUM DUADIC AND AFFINE-INVARIANT CODES

Kenza Guenda

Published Feb 1, 2009• 11 citations
Publisher

Abstract

We construct new families of duadic codes and derive from them degenerate quantum codes. This provides a solution to a question raised by Aly, Klappenecker and Sarvepalli on the existence of quantum codes from duadic codes when the multiplicative order of q modulo n is even. We also characterize the affine-invariant maximal extended cyclic codes. Then by the CSS construction, we give a family of quantum code.

DOI
10.1142/s0219749909004979

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We construct new families of duadic codes and derive from them degenerate quantum codes. This provides a solution to a question raised by Aly, Klappenecker and Sarvepalli on the existence of quantum codes from duadic codes when the multiplicative order of q modulo n is even. We also characterize the affine-invariant maximal extended cyclic codes. Then by the CSS construction, we give a family of quantum code.
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
In this paper, we use 2-cyclotomic cosets of modulo n and generator polynomials to describe binary cyclic codes of length N=2αn with n odd. We discuss the conditions under which two cyclic codes [Formula: see text] and [Formula: see text] can be used to construct quantum codes by CSS construction or Steane's construction. Using the results of Chen, Promhouse and Tavares, and Castagnoli et al., we study the quantum codes that can be constructed from binary cyclic codes of length N=2αn with n odd and n≤99, and α≤2. We find that except the quantum codes constructed by Steane, there are also some very interesting quantum codes constructed from repeated-root cyclic codes, and some of the quantum codes constructed by Steane can be improved.
In this paper we extend to asymmetric quantum error-correcting codes (AQECC) the construction methods, namely: puncturing, extending, expanding, direct sum and the (u|u + v) construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes, quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and affine-invariant codes are constructed.