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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

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.  Google Scholar

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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.  Google Scholar

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M. Deza, Une propriété extrémale des plans projectifs finis dans une classe de codes équidistants, Discrete Math., 6 (1973), 343-352.  Google Scholar

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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. Google Scholar

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P. Kaski and O. Pottonen, libexact user's guide, version 1.0, HIIT Technical Reports 2008-1, Helsinki, 2008. Google Scholar

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D. S. Krotov, On diameter perfect constant-weight ternary codes, Discrete Math., 308 (2008), 3104-3114. doi: 10.1016/j.disc.2007.08.037.  Google Scholar

[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.  Google Scholar

[10]

P. R. J. Östergård and M. Svanström, Ternary constant weight codes, Electr. J. Combin., 9(1) (2002), #R41.  Google Scholar

[11]

M. Svanström, A class of perfect ternary constant-weight codes, Des. Codes Crypt., 18 (1999), 223-229. doi: 10.1023/A:1008361925021.  Google Scholar

[12]

M. Svanström, Ternary Codes with Weight Constraints, Ph.D thesis, Linköping Univ., 1999. Google Scholar

[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.  Google Scholar

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J. van Lint and L. Tolhuizen, On perfect ternary constant weight codes, Des. Codes Crypt., 18 (1999), 231-234. doi: 10.1023/A:1008314009092.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Deza, Une propriété extrémale des plans projectifs finis dans une classe de codes équidistants, Discrete Math., 6 (1973), 343-352.  Google Scholar

[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. Google Scholar

[7]

P. Kaski and O. Pottonen, libexact user's guide, version 1.0, HIIT Technical Reports 2008-1, Helsinki, 2008. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. R. J. Östergård and M. Svanström, Ternary constant weight codes, Electr. J. Combin., 9(1) (2002), #R41.  Google Scholar

[11]

M. Svanström, A class of perfect ternary constant-weight codes, Des. Codes Crypt., 18 (1999), 223-229. doi: 10.1023/A:1008361925021.  Google Scholar

[12]

M. Svanström, Ternary Codes with Weight Constraints, Ph.D thesis, Linköping Univ., 1999. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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