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

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

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

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.
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.   Google Scholar

[2]

K. Betsumiya, On the classification of type II codes over $\mathbbF_{2^r}$ with binary length 32,, preprint., ().   Google Scholar

[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.  doi: 10.1109/18.945245.  Google Scholar

[4]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[5]

T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes,, Adv. Math. Commun., 3 (2009), 363.  doi: 10.3934/amc.2009.3.363.  Google Scholar

[6]

P. Gaborit, V. Pless, P. Solé and O. Atkin, Type II codes over $\mathbbF_4$,, Finite Fields Appl., 8 (2002), 171.  doi: 10.1006/ffta.2001.0333.  Google Scholar

[7]

A. Günther, Codes und Invariantentheorie (in German),, Diploma thesis, (2006).   Google Scholar

[8]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge University Press, (2003).  doi: 10.1017/CBO9780511807077.  Google Scholar

[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.  doi: 10.1016/j.ffa.2003.12.001.  Google Scholar

[10]

H.-G. Quebbemann, On even codes,, Discrete Math., 98 (1991), 29.  doi: 10.1016/0012-365X(91)90410-4.  Google Scholar

[11]

The Sage Development Team, Sage Mathematics Software, Version 6.4.1,, , ().   Google Scholar

show all references

References:
[1]

E. F. Assmus, Jr. and H. F. Mattson, Jr., New $5$-designs,, J. Combin. Theory, 6 (1969), 122.   Google Scholar

[2]

K. Betsumiya, On the classification of type II codes over $\mathbbF_{2^r}$ with binary length 32,, preprint., ().   Google Scholar

[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.  doi: 10.1109/18.945245.  Google Scholar

[4]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[5]

T. Feulner, The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes,, Adv. Math. Commun., 3 (2009), 363.  doi: 10.3934/amc.2009.3.363.  Google Scholar

[6]

P. Gaborit, V. Pless, P. Solé and O. Atkin, Type II codes over $\mathbbF_4$,, Finite Fields Appl., 8 (2002), 171.  doi: 10.1006/ffta.2001.0333.  Google Scholar

[7]

A. Günther, Codes und Invariantentheorie (in German),, Diploma thesis, (2006).   Google Scholar

[8]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes,, Cambridge University Press, (2003).  doi: 10.1017/CBO9780511807077.  Google Scholar

[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.  doi: 10.1016/j.ffa.2003.12.001.  Google Scholar

[10]

H.-G. Quebbemann, On even codes,, Discrete Math., 98 (1991), 29.  doi: 10.1016/0012-365X(91)90410-4.  Google Scholar

[11]

The Sage Development Team, Sage Mathematics Software, Version 6.4.1,, , ().   Google Scholar

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