# 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:
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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. doi: 10.1006/jsco.1996.0125. Google Scholar [2] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Applied Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. Google Scholar [3] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings,, in, (2009), 38. doi: 10.1007/978-3-642-10868-6_3. Google Scholar [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. doi: 10.1109/18.681315. Google Scholar [5] M. F. Ezerman, M. Grassl and P. Solé, The weights in MDS codes,, IEEE Trans. Inform. Theory, 57 (2011), 392. doi: 10.1109/TIT.2010.2090246. Google Scholar [6] M. F. Ezerman, S. Ling and P. Solé, Additive asymmetric quantum codes,, preprint, (). Google Scholar [7] K. Feng, S. Ling and C. Xing, Asymptotic bounds on quantum codes from algebraic geometry codes,, IEEE Trans. Inform. Theory, 52 (2006), 986. doi: 10.1109/TIT.2005.862086. Google Scholar [8] E. M. Gabidulin, Theory of codes with maximum rank distance,, Probl. Peredach. Inform. (in Russian), 21 (1985), 3. Google Scholar [9] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at \url{http://www.codetables.de}, (). Google Scholar [10] W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes,'', Cambridge University Press, (2003). Google Scholar [11] J. L. Kim and V. Pless, Designs in additive codes over $GF(4)$,, Des. Codes Crypt., 30 (2003), 187. doi: 10.1023/A:1025484821641. Google Scholar [12] S. Ling and C. P. Xing, "Coding Theory. A First Course,'', Cambridge University Press, (2004). doi: 10.1017/CBO9780511755279. Google Scholar [13] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'', North-Holland, (1977). Google Scholar [14] G. Nebe, E. M. Rains and N. J. A. Sloane, "Self-Dual Codes and Invariant Theory,'', Springer-Verlag, (2006). Google Scholar [15] E. M. Rains and N. J. A. Sloane, Self-dual codes,, in, (1998), 177. Google Scholar [16] P. K. Sarvepalli, A. Klappenecker and M. Rötteler, Asymmetric quantum codes: constructions, bounds and performance,, Proc. Royal Soc. A, 465 (2009), 1645. doi: 10.1098/rspa.2008.0439. Google Scholar [17] L. Wang, K. Feng, S. Ling and C. Xing, Asymmetric quantum codes: characterization and constructions,, IEEE Trans. Inform. Theory, 56 (2010), 2938. doi: 10.1109/TIT.2010.2046221. Google Scholar
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