Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound

It is known that quantum error correction can be achieved using classical binary codes or additive codes over F/sub 4/. Asymptotically good quantum codes have been constructed from algebraic-geometry codes and a bound on (/spl delta/, R) was computed from the Tsfasman-Vladut-Zink bound of the theory of classical algebraic-geometry codes. In this correspondence, by the use of a concatenation technique we construct a family of asymptotically good quantum codes exceeding the bound in a small interval.

[1]  N. Sloane,et al.  Quantum Error Correction Via Codes Over GF , 1998 .

[2]  Chaoping Xing,et al.  A generalization of algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[3]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  Jennifer Seberry,et al.  On inequivalent weighing matrices , 1986 .

[5]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[6]  Gérard D. Cohen,et al.  On binary constructions of quantum codes , 1999, IEEE Trans. Inf. Theory.

[7]  N. J. A. Sloane,et al.  Self-Dual Codes over GF(5) , 1982, J. Comb. Theory, Ser. A.

[8]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[9]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[10]  Masaaki Harada,et al.  New Optimal Self-Dual Codes over GF(7) , 1999, Graphs Comb..

[11]  S. Litsyn,et al.  Asymptotically Good Quantum Codes , 2000, quant-ph/0006061.

[12]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[13]  H. Stichtenoth,et al.  On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields , 1996 .

[14]  Vladimir D. Tonchev,et al.  Self-dual codes over GF(7) , 1987, IEEE Trans. Inf. Theory.

[15]  Peter J. Cameron,et al.  Designs, graphs, codes, and their links , 1991 .

[16]  Andrew M. Steane.  Enlargement of Calderbank-Shor-Steane quantum codes , 1999, IEEE Trans. Inf. Theory.

[17]  A. Brouwer.  Bounds on the size of linear codes , 1998 .

[18]  Markus Grassl,et al.  Quantum Reed-Solomon Codes , 1999, AAECC.

[19]  Hao Chen.  Some good quantum error-correcting codes from algebraic-Geometric codes , 2001, IEEE Trans. Inf. Theory.

[20]  Gerard F. M. Beenker,et al.  A note on extended quadratic residue codes over GF(9) and their ternary images , 1984, IEEE Trans. Inf. Theory.

[21]  J. H. Lint,et al.  Algebraic-geometric codes , 1992 .