Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes

In this paper, we construct new families of classical convolutional codes (CCC’s) and new families of quantum convolutional codes (QCC’s). The CCC’s are derived from (block) algebraic geometry (AG) codes. Furthermore, new families of CCC’s are constructed by applying the techniques of puncturing, extending, expanding, and by the direct product code construction applied to AG codes. In addition, utilizing the new CCC’s constructed here, we obtain new families of QCC’s. The parameters of these new codes are good. More precisely, in the classical case, a family of almost near maximum distance separable (MDS) codes is presented; in the quantum case, we construct a family of MDS (optimal) quantum convolutional codes.

[1]  Joachim Rosenthal,et al.  Maximum Distance Separable Convolutional Codes , 1999, Applicable Algebra in Engineering, Communication and Computing.

[2]  Giuliano G. La Guardia,et al.  On Classical and Quantum MDS-Convolutional BCH Codes , 2012, IEEE Transactions on Information Theory.

[3]  H. Stichtenoth.  Self-dual Goppa codes , 1988 .

[4]  Henning Stichtenoth,et al.  Constructing codes from algebraic curves , 1999, IEEE Trans. Inf. Theory.

[5]  Heide Gluesing-Luerssen,et al.  A MacWilliams Identity for Convolutional Codes: The General Case , 2009, IEEE Transactions on Information Theory.

[6]  H. Niederreiter,et al.  Towers of Global Function Fields with Asymptotically Many Rational Places and an Improvement on the Gilbert ‐ Varshamov Bound , 1998 .

[7]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

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

[9]  Jean-Pierre Tillich,et al.  Description of a quantum convolutional code. , 2003, Physical review letters.

[10]  Heide Gluesing-Luerssen,et al.  A matrix ring description for cyclic convolutional codes , 2008, Adv. Math. Commun..

[11]  Philippe Piret,et al.  Convolutional Codes: An Algebraic Approach , 1988 .

[12]  Joachim Rosenthal,et al.  Constructions of MDS-convolutional codes , 2001, IEEE Trans. Inf. Theory.

[13]  Chaoping Xing,et al.  Euclidean and Hermitian Self-Orthogonal Algebraic Geometry Codes and Their Application to Quantum Codes , 2012, IEEE Transactions on Information Theory.

[14]  Lingfei Jin.  Quantum Stabilizer Codes From Maximal Curves , 2014, IEEE Transactions on Information Theory.

[15]  Lin-nan Lee,et al.  Short unit-memory byte-oriented binary convolutional codes having maximal free distance (Corresp.) , 1976, IEEE Trans. Inf. Theory.

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

[17]  F. J. Plaza-Martin,et al.  On the Construction of 1-D MDS Convolutional Goppa Codes , 2013, IEEE Transactions on Information Theory.

[18]  E. Knill,et al.  Theory of quantum error-correcting codes , 1997 .

[19]  Giuliano G. La Guardia.  On optimal constacyclic codes , 2013, ArXiv.

[20]  H. Gluesing-Luerssen,et al.  On the MacWilliams Identity for Convolutional Codes , 2006, IEEE Transactions on Information Theory.

[21]  Carlos Munuera,et al.  Quantum error-correcting codes from algebraic geometry codes of Castle type , 2016, Quantum Inf. Process..

[22]  Ruud Pellikaan,et al.  Which linear codes are algebraic-geometric? , 1991, IEEE Trans. Inf. Theory.

[23]  José Ignacio Iglesias Curto,et al.  Convolutional Goppa codes defined on fibrations , 2012, Applicable Algebra in Engineering, Communication and Computing.

[24]  Giuliano G. La Guardia.  Nonbinary convolutional codes derived from group character codes , 2013, Discret. Math..

[25]  Chaoping Xing,et al.  Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes , 2010, IEEE Transactions on Information Theory.

[26]  G. David Forney,et al.  Convolutional codes I: Algebraic structure , 1970, IEEE Trans. Inf. Theory.

[27]  Pradeep Kiran Sarvepalli,et al.  Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes , 2007, ArXiv.

[28]  Henning Stichtenoth,et al.  Group codes on certain algebraic curves with many rational points , 1990, Applicable Algebra in Engineering, Communication and Computing.

[29]  Giuliano G. La Guardia,et al.  On Negacyclic MDS-Convolutional Codes , 2013, ArXiv.

[30]  Heide Gluesing-Luerssen.  On the weight distribution of convolutional codes , 2005, ArXiv.

[31]  H. Ollivier,et al.  Quantum convolutional codes: fundamentals , 2004 .

[32]  M. Tsfasman,et al.  Algebraic Geometric Codes: Basic Notions , 2007 .

[33]  J. M. Muñoz Porras,et al.  Convolutional Codes of Goppa Type , 2003, Applicable Algebra in Engineering, Communication and Computing.

[34]  Giuliano Gadioli La Guardia,et al.  Good and asymptotically good quantum codes derived from algebraic geometry , 2016, Quantum Inf. Process..

[35]  Pradeep Kiran Sarvepalli,et al.  Quantum Convolutional BCH Codes , 2007, 2007 10th Canadian Workshop on Information Theory (CWIT).