Non-existence of a ternary constant weight (16,5,15;2048) diameter perfect code

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May 2016, 10(2): 393-399. doi: 10.3934/amc.2016013

Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code

Denis S. Krotov ^1, , Patric R. J. Östergård ^2, and Olli Pottonen ^3,

1.	Sobolev Institute of Mathematics, Mechanics and Mathematics Department, Novosibirsk State University, Novosibirsk, Russian Federation
2.	Department of Communications and Networking, School of Electrical Engineering, Aalto University, P.O. Box 13000, 00076 Aalto, Finland
3.	School of Mathematics and Physics, The University of Queensland, Brisbane, Australia

Received August 2014 Revised December 2014 Published April 2016

Abstract
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Ternary constant weight codes of length $n=2^m$, weight $n-1$, cardinality $2^n$ and distance $5$ are known to exist for every $m$ for which there exists an APN permutation of order $2^m$, that is, at least for all odd $m \geq 3$ and for $m=6$. We show the non-existence of such codes for $m=4$ and prove that any codes with the parameters above are diameter perfect.

Keywords: diameter perfect code., Constant weight code.

Mathematics Subject Classification: Primary: 94B2.

Citation: Denis S. Krotov, Patric R. J. Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013

References:

[1]	R. Ahlswede, H. K. Aydinian and L. H. Khachatrian, On perfect codes and related concepts, Des. Codes Crypt., 22 (2001), 221-237. doi: 10.1023/A:1008394205999.
[2]	R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, Eur. J. Combin., 18 (1997), 125-136. doi: 10.1006/eujc.1995.0092.
[3]	R. Ahlswede and L. H. Khachatrian, The diametric theorem in Hamming spaces-optimal anticodes, Adv. Appl. Math., 20 (1998), 429-449. doi: 10.1006/aama.1998.0588.
[4]	K. A. Browning, J. F. Dillon, M. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, in Finite Fields: Theory and Applications, Amer. Math. Soc., Providence, 2010, 33-42. doi: 10.1090/conm/518/10194.
[5]	M. Deza, Une propriété extrémale des plans projectifs finis dans une classe de codes équidistants, Discrete Math., 6 (1973), 343-352.
[6]	T. Junttila and P. Kaski, Engineering an efficient canonical labeling tool for large and sparse graphs, in Proc. 9th Workshop Algor. Engin. Exper., Soc. Industr. Appl. Math., Philadelphia, 2007, 135-149.
[7]	P. Kaski and O. Pottonen, libexact user's guide, version 1.0, HIIT Technical Reports 2008-1, Helsinki, 2008.
[8]	D. S. Krotov, On diameter perfect constant-weight ternary codes, Discrete Math., 308 (2008), 3104-3114. doi: 10.1016/j.disc.2007.08.037.
[9]	P. R. J. Östergård and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I-classification, IEEE Trans. Inf. Theory, 55 (2009), 4657-4660. doi: 10.1109/TIT.2009.2027525.
[10]	P. R. J. Östergård and M. Svanström, Ternary constant weight codes, Electr. J. Combin., 9(1) (2002), #R41.
[11]	M. Svanström, A class of perfect ternary constant-weight codes, Des. Codes Crypt., 18 (1999), 223-229. doi: 10.1023/A:1008361925021.
[12]	M. Svanström, Ternary Codes with Weight Constraints, Ph.D thesis, Linköping Univ., 1999.
[13]	H. Tanaka, Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs, J. Combin. Theory Ser. A, 113 (2006), 903-910. doi: 10.1016/j.jcta.2005.08.006.
[14]	J. van Lint and L. Tolhuizen, On perfect ternary constant weight codes, Des. Codes Crypt., 18 (1999), 231-234. doi: 10.1023/A:1008314009092.

show all references

References:

[1]	R. Ahlswede, H. K. Aydinian and L. H. Khachatrian, On perfect codes and related concepts, Des. Codes Crypt., 22 (2001), 221-237. doi: 10.1023/A:1008394205999.
[2]	R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, Eur. J. Combin., 18 (1997), 125-136. doi: 10.1006/eujc.1995.0092.
[3]	R. Ahlswede and L. H. Khachatrian, The diametric theorem in Hamming spaces-optimal anticodes, Adv. Appl. Math., 20 (1998), 429-449. doi: 10.1006/aama.1998.0588.
[4]	K. A. Browning, J. F. Dillon, M. T. McQuistan and A. J. Wolfe, An APN permutation in dimension six, in Finite Fields: Theory and Applications, Amer. Math. Soc., Providence, 2010, 33-42. doi: 10.1090/conm/518/10194.
[5]	M. Deza, Une propriété extrémale des plans projectifs finis dans une classe de codes équidistants, Discrete Math., 6 (1973), 343-352.
[6]	T. Junttila and P. Kaski, Engineering an efficient canonical labeling tool for large and sparse graphs, in Proc. 9th Workshop Algor. Engin. Exper., Soc. Industr. Appl. Math., Philadelphia, 2007, 135-149.
[7]	P. Kaski and O. Pottonen, libexact user's guide, version 1.0, HIIT Technical Reports 2008-1, Helsinki, 2008.
[8]	D. S. Krotov, On diameter perfect constant-weight ternary codes, Discrete Math., 308 (2008), 3104-3114. doi: 10.1016/j.disc.2007.08.037.
[9]	P. R. J. Östergård and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I-classification, IEEE Trans. Inf. Theory, 55 (2009), 4657-4660. doi: 10.1109/TIT.2009.2027525.
[10]	P. R. J. Östergård and M. Svanström, Ternary constant weight codes, Electr. J. Combin., 9(1) (2002), #R41.
[11]	M. Svanström, A class of perfect ternary constant-weight codes, Des. Codes Crypt., 18 (1999), 223-229. doi: 10.1023/A:1008361925021.
[12]	M. Svanström, Ternary Codes with Weight Constraints, Ph.D thesis, Linköping Univ., 1999.
[13]	H. Tanaka, Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs, J. Combin. Theory Ser. A, 113 (2006), 903-910. doi: 10.1016/j.jcta.2005.08.006.
[14]	J. van Lint and L. Tolhuizen, On perfect ternary constant weight codes, Des. Codes Crypt., 18 (1999), 231-234. doi: 10.1023/A:1008314009092.

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