On Negacyclic MDS-Convolutional Codes

New families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper. This optimality is in the sense that they attain the classical (quantum) generalized Singleton bound. The constructions presented in this paper are performed algebraically and not by computational search.

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

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

[3]  Dwijendra K. Ray-Chaudhuri,et al.  The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes , 2001, Des. Codes Cryptogr..

[4]  Shixin Zhu,et al.  New Quantum MDS Codes From Negacyclic Codes , 2013, IEEE Transactions on Information Theory.

[5]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[6]  José Ignacio Iglesias Curto,et al.  Generalized AG convolutional codes , 2009, Adv. Math. Commun..

[7]  Shixin Zhu,et al.  Quantum negacyclic codes , 2013 .

[8]  Joachim Rosenthal,et al.  BCH convolutional codes , 1999, IEEE Trans. Inf. Theory.

[9]  Heide Gluesing-Luerssen,et al.  On Doubly-Cyclic Convolutional Codes , 2006, Applicable Algebra in Engineering, Communication and Computing.

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

[11]  Madhu Raka,et al.  A class of constacyclic codes over a finite field-II , 2012, Indian Journal of Pure and Applied Mathematics.

[12]  Philippe Piret A convolutional equivalent to Reed-Solomon codes , 1988 .

[13]  Joachim Rosenthal,et al.  Convolutional codes with maximum distance profile , 2003, Syst. Control. Lett..

[14]  Martin Rötteler,et al.  Constructions of Quantum Convolutional Codes , 2007, 2007 IEEE International Symposium on Information Theory.

[15]  Reginaldo Palazzo Júnior,et al.  A concatenated [(4, 1, 3)] quantum convolutional code , 2004, Information Theory Workshop.

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

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

[19]  Joachim Rosenthal,et al.  Strongly-MDS convolutional codes , 2003, IEEE Transactions on Information Theory.

[20]  Thomas Blackford Negacyclic duadic codes , 2008, Finite Fields Their Appl..

[21]  Martin Rötteler,et al.  Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes , 2006, 2006 IEEE International Symposium on Information Theory.

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

[23]  Rolf Johannesson,et al.  Fundamentals of Convolutional Coding , 1999 .

[24]  Heide Gluesing-Luerssen,et al.  Distance Bounds for Convolutional Codes and Some Optimal Codes , 2003 .

[25]  Markus Grassl,et al.  Convolutional and Tail-Biting Quantum Error-Correcting Codes , 2005, IEEE Transactions on Information Theory.

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

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

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

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

[30]  Av . Van Becelaere A CONVOLUTIONAL EQUIVALENT TO REED- SOLOMON CODES , 1988 .

[31]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[32]  Santosh Kumar,et al.  Nonbinary Stabilizer Codes Over Finite Fields , 2005, IEEE Transactions on Information Theory.

[33]  Joan-Josep Climent,et al.  Linear system modelization of concatenated block and convolutional codes , 2008 .

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

[35]  Saroj Rani,et al.  On constacyclic codes over finite fields , 2015, Cryptography and Communications.

[36]  Kjell Jørgen Hole On classes of convolutional codes that are not asymptotically catastrophic , 2000, IEEE Trans. Inf. Theory.

[37]  Giuliano G. La Guardia,et al.  New Quantum MDS Codes , 2011, IEEE Transactions on Information Theory.

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

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

[40]  G. L. Guardia Constructions of new families of nonbinary quantum codes , 2009 .

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

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

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

[44]  Elwyn R. Berlekamp Negacyclic codes for the Lee metric , 1966 .

[45]  Dilip V. Sarwate,et al.  Pseudocyclic maximum- distance-separable codes , 1990, IEEE Trans. Inf. Theory.

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

[47]  Martin Rötteler,et al.  Quantum block and convolutional codes from self-orthogonal product codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..