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Codes over local rings of order 16 and binary codes
Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code
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 |
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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