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. 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

show all references

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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