# American Institute of Mathematical Sciences

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
 [1] Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251 [2] Masaaki Harada. New doubly even self-dual codes having minimum weight 20. Advances in Mathematics of Communications, 2020, 14 (1) : 89-96. doi: 10.3934/amc.2020007 [3] Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433 [4] Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261 [5] Masaaki Harada, Katsushi Waki. New extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2009, 3 (4) : 311-316. doi: 10.3934/amc.2009.3.311 [6] Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031 [7] Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229 [8] Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267 [9] Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047 [10] Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002 [11] Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23 [12] Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393 [13] Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65 [14] Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415 [15] Jongmin Han, Juhee Sohn. On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 819-839. doi: 10.3934/dcds.2019034 [16] Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011 [17] Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020023 [18] Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219 [19] Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425 [20] Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437

2018 Impact Factor: 0.879