A polynomial-time construction of self-orthogonal codes and applications to quantum error correction

A polynomial-time construction of a sequence of self-orthogonal geometric Goppa codes attaining the Tsfasman-Vlăduţ-Zink (TVZ) bound is presented. The issue of constructing such a code sequence was addressed in a context of constructing quantum error-correcting codes (Ashikhmin et al., 2001). Naturally, the obtained construction has implications on quantum error-correcting codes. In particular, the best known asymptotic lower bounds on the largest minimum distance of polynomially constructible quantum error-correcting codes are improved.

[1]  H. Stichtenoth,et al.  A low complexity algorithm for the construction of algebraic geometric codes better than the Gilbert-Varshamov bound , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

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

[3]  Tom Høholdt,et al.  An explicit construction of a sequence of codes attaining the Tsfasman-Vladut-Zink bound: The first steps , 1997, IEEE Trans. Inf. Theory.

[4]  H. Fujita.  Several Classes of Concatenated Quantum Codes: Constructions and Bounds , 2006, quant-ph/0608063.

[5]  V. D. Goppa.  Codes on Algebraic Curves , 1981 .

[6]  M. Hamada.  Concatenated Quantum Codes Constructible in Polynomial Time: Efficient Decoding and Error Correction , 2008, IEEE Transactions on Information Theory.

[7]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[8]  Chaoping Xing,et al.  Asymptotic bounds on quantum codes from algebraic geometry codes , 2006, IEEE Transactions on Information Theory.

[9]  Henning Stichtenoth,et al.  Transitive and self-dual codes attaining the Tsfasman-Vla/spl breve/dut$80-Zink bound , 2006, IEEE Transactions on Information Theory.

[10]  Fujita Hachiro.  Several Classes of Concatenated Quantum Codes: Constructions and Bounds , 2006 .

[11]  Kenneth W. Shum,et al.  A low-complexity construction of algebraic geometric codes better than the gilbert-varshamov bound , 2000 .

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

[13]  Alexei E. Ashikhmin,et al.  Nonbinary quantum stabilizer codes , 2001, IEEE Trans. Inf. Theory.

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

[15]  Henning Stichtenoth.  Transitive and Self-dual Codes Attaining the Tsfasman-Vladut-Zink Bound , 2005 .

[16]  M. Hamada.  Information rates achievable with algebraic codes on quantum discrete memoryless channels , 2005, IEEE Transactions on Information Theory.

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

[18]  Some Artin–Schreier Towers Are Easy , 2005 .

[19]  Hao Chen,et al.  Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound , 2001, IEEE Trans. Inf. Theory.

[20]  Ryutaroh Matsumoto,et al.  Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[21]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[22]  Douglas A. Leonard,et al.  Finding the defining functions for one-point algebraic-geometry codes , 2001, IEEE Trans. Inf. Theory.

[23]  R. Cleve,et al.  Efficient computations of encodings for quantum error correction , 1996, quant-ph/9607030.