# American Institute of Mathematical Sciences

February  2011, 5(1): 41-57. doi: 10.3934/amc.2011.5.41

## From skew-cyclic codes to asymmetric quantum codes

 1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore, Singapore 2 Centre National de la Recherche Scienti, Telecom-ParisTech, Dept Comelec, 46 rue Barrault, 75634 Paris, France 3 Institut Préparatoire aux Études d'Ingénieurs El Manar, Campus Universitaire El Manar, Tunis, Tunisia

Received  May 2010 Published  February 2011

We introduce an additive but not $\mathbb F$4-linear map $S$ from $\mathbb F$4n to $\mathbb F$42n and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]$4-code, then $S(C)$ is an additive $(2n,2$2k,$2d)$4-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)$4-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.
Citation: Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41
##### References:
 [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. [2] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Applied Algebra Engin. Commun. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. [3] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, in "Proceedings of the 12th IMA Conference on Cryptography and Coding,'' Cirencester, (2009), 38-55. doi: 10.1007/978-3-642-10868-6_3. [4] A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315. [5] M. F. Ezerman, M. Grassl and P. Solé, The weights in MDS codes, IEEE Trans. Inform. Theory, 57 (2011), 392-396. doi: 10.1109/TIT.2010.2090246. [6] M. F. Ezerman, S. Ling and P. Solé, Additive asymmetric quantum codes,, preprint, (). [7] K. Feng, S. Ling and C. Xing, Asymptotic bounds on quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 52 (2006), 986-991. doi: 10.1109/TIT.2005.862086. [8] E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform. (in Russian), 21 (1985), 3-16; English translation, 1-12. [9] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, (). [10] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. [11] J. L. Kim and V. Pless, Designs in additive codes over $GF(4)$, Des. Codes Crypt., 30 (2003), 187-199. doi: 10.1023/A:1025484821641. [12] S. Ling and C. P. Xing, "Coding Theory. A First Course,'' Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755279. [13] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977. [14] G. Nebe, E. M. Rains and N. J. A. Sloane, "Self-Dual Codes and Invariant Theory,'' Springer-Verlag, Berlin, Heidelberg, 2006. [15] E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory I'' (eds. V.S. Pless and W.C. Huffman), Elsevier, (1998), 177-294. [16] P. K. Sarvepalli, A. Klappenecker and M. Rötteler, Asymmetric quantum codes: constructions, bounds and performance, Proc. Royal Soc. A, 465 (2009), 1645-1672. doi: 10.1098/rspa.2008.0439. [17] L. Wang, K. Feng, S. Ling and C. Xing, Asymmetric quantum codes: characterization and constructions, IEEE Trans. Inform. Theory, 56 (2010), 2938-2945. doi: 10.1109/TIT.2010.2046221.

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##### References:
 [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. [2] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Applied Algebra Engin. Commun. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. [3] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, in "Proceedings of the 12th IMA Conference on Cryptography and Coding,'' Cirencester, (2009), 38-55. doi: 10.1007/978-3-642-10868-6_3. [4] A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $GF(4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315. [5] M. F. Ezerman, M. Grassl and P. Solé, The weights in MDS codes, IEEE Trans. Inform. Theory, 57 (2011), 392-396. doi: 10.1109/TIT.2010.2090246. [6] M. F. Ezerman, S. Ling and P. Solé, Additive asymmetric quantum codes,, preprint, (). [7] K. Feng, S. Ling and C. Xing, Asymptotic bounds on quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 52 (2006), 986-991. doi: 10.1109/TIT.2005.862086. [8] E. M. Gabidulin, Theory of codes with maximum rank distance, Probl. Peredach. Inform. (in Russian), 21 (1985), 3-16; English translation, 1-12. [9] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, (). [10] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'' Cambridge University Press, Cambridge, 2003. [11] J. L. Kim and V. Pless, Designs in additive codes over $GF(4)$, Des. Codes Crypt., 30 (2003), 187-199. doi: 10.1023/A:1025484821641. [12] S. Ling and C. P. Xing, "Coding Theory. A First Course,'' Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511755279. [13] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1977. [14] G. Nebe, E. M. Rains and N. J. A. Sloane, "Self-Dual Codes and Invariant Theory,'' Springer-Verlag, Berlin, Heidelberg, 2006. [15] E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory I'' (eds. V.S. Pless and W.C. Huffman), Elsevier, (1998), 177-294. [16] P. K. Sarvepalli, A. Klappenecker and M. Rötteler, Asymmetric quantum codes: constructions, bounds and performance, Proc. Royal Soc. A, 465 (2009), 1645-1672. doi: 10.1098/rspa.2008.0439. [17] L. Wang, K. Feng, S. Ling and C. Xing, Asymmetric quantum codes: characterization and constructions, IEEE Trans. Inform. Theory, 56 (2010), 2938-2945. doi: 10.1109/TIT.2010.2046221.
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