There is no [24,12,9] doubly-even self-dual code over \mathbb F_4

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August 2016, 10(3): 583-588. doi: 10.3934/amc.2016027

There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$

Sihuang Hu ^1, and Gabriele Nebe ^2,

1.	Department of EE-Systems, Tel Aviv University, Tel Aviv, Israel
2.	Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen

Received May 2015 Revised October 2015 Published August 2016

Abstract
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We show that there is no $[24,12,9]$ doubly-even self-dual code over $\mathbb{F}_4$ by attempting to construct the generator matrix of this code directly.

Keywords: Doubly-even, self-dual..

Mathematics Subject Classification: Primary: 94B0.

Citation: Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027

References:

[1]	E. F. Assmus, Jr. and H. F. Mattson, Jr., New $5$-designs, J. Combin. Theory, 6 (1969), 122-151.
[2]	K. Betsumiya, On the classification of type II codes over $\mathbbF_{2^r}$ with binary length 32, preprint.
[3]	K. Betsumiya, T. A. Gulliver, M. Harada and A. Munemasa, On type II codes over $\mathbbF_4$, IEEE Trans. Inform. Theory, 47 (2001), 2242-2248. doi: 10.1109/18.945245.
[4]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[5]	T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Adv. Math. Commun., 3 (2009), 363-383. doi: 10.3934/amc.2009.3.363.
[6]	P. Gaborit, V. Pless, P. Solé and O. Atkin, Type II codes over $\mathbbF_4$, Finite Fields Appl., 8 (2002), 171-183. doi: 10.1006/ffta.2001.0333.
[7]	A. Günther, Codes und Invariantentheorie (in German), Diploma thesis, RWTH Aachen, 2006. Available online at http://www.math.rwth-aachen.de/~Gabriele.Nebe/dipl/guenther.pdf
[8]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.
[9]	G. Nebe, H.-G. Quebbemann, E. M. Rains and N. J. A. Sloane, Complete weight enumerators of generalized doubly-even self-dual codes, Finite Fields Appl., 10 (2004), 540-550. doi: 10.1016/j.ffa.2003.12.001.
[10]	H.-G. Quebbemann, On even codes, Discrete Math., 98 (1991), 29-34. doi: 10.1016/0012-365X(91)90410-4.
[11]	The Sage Development Team, Sage Mathematics Software, Version 6.4.1, http://www.sagemath.org

show all references

References:

[1]	E. F. Assmus, Jr. and H. F. Mattson, Jr., New $5$-designs, J. Combin. Theory, 6 (1969), 122-151.
[2]	K. Betsumiya, On the classification of type II codes over $\mathbbF_{2^r}$ with binary length 32, preprint.
[3]	K. Betsumiya, T. A. Gulliver, M. Harada and A. Munemasa, On type II codes over $\mathbbF_4$, IEEE Trans. Inform. Theory, 47 (2001), 2242-2248. doi: 10.1109/18.945245.
[4]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[5]	T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes, Adv. Math. Commun., 3 (2009), 363-383. doi: 10.3934/amc.2009.3.363.
[6]	P. Gaborit, V. Pless, P. Solé and O. Atkin, Type II codes over $\mathbbF_4$, Finite Fields Appl., 8 (2002), 171-183. doi: 10.1006/ffta.2001.0333.
[7]	A. Günther, Codes und Invariantentheorie (in German), Diploma thesis, RWTH Aachen, 2006. Available online at http://www.math.rwth-aachen.de/~Gabriele.Nebe/dipl/guenther.pdf
[8]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.
[9]	G. Nebe, H.-G. Quebbemann, E. M. Rains and N. J. A. Sloane, Complete weight enumerators of generalized doubly-even self-dual codes, Finite Fields Appl., 10 (2004), 540-550. doi: 10.1016/j.ffa.2003.12.001.
[10]	H.-G. Quebbemann, On even codes, Discrete Math., 98 (1991), 29-34. doi: 10.1016/0012-365X(91)90410-4.
[11]	The Sage Development Team, Sage Mathematics Software, Version 6.4.1, http://www.sagemath.org

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