Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7

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February 2014, 8(1): 73-81. doi: 10.3934/amc.2014.8.73

Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7

Nikolay Yankov ^1,

1.	Faculty of Mathematics and Informatics, Konstantin Preslavski University of Shumen, Shumen, 9712, Bulgaria

Received February 2013 Revised June 2013 Published January 2014

Abstract
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This paper studies and classifies all binary self-dual $[62, 31, 12]$ and $[64, 32, 12]$ codes having an automorphism of order 7 with 8 cycles. This classification is done by applying a method for constructing binary self-dual codes with an automorphism of odd prime order $p$. There are exactly 8 inequivalent binary self-dual $[62, 31, 12]$ codes with an automorphism of type $7-(8,6)$. As for binary $[64,32,12]$ self-dual codes with an automorphism of type $7-(8,8)$ there are 44465 doubly-even and 557 singly-even such codes. Some of the constructed singly-even codes for both lengths have weight enumerators for which the existence was not known before.

Keywords: self-dual codes., Automorphisms.

Mathematics Subject Classification: Primary: 11T71; Secondary: 94B0.

Citation: Nikolay Yankov. Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7. Advances in Mathematics of Communications, 2014, 8 (1) : 73-81. doi: 10.3934/amc.2014.8.73

References:

[1]	I. Bouyukliev, About the code equivalence, in Advances in Coding Theory and Cryptography, World Scientific Publishing, 2007. doi: 10.1142/9789812772022_0009.
[2]	S. Bouyuklieva, N. Yankov and J.-L. Kim, Classification of binary self-dual [48, 24, 10] codes with an automorphism of odd prime order, Finite Fields Appl., 18 (2012), 1104-1113. doi: 10.1016/j.ffa.2012.08.002.
[3]	S. Bouyuklieva, N. Yankov and R. Russeva, Classification of the binary self-dual codes having an automorphism of order 3, Finite Fields Appl., 13 (2007), 605-615. doi: 10.1016/j.ffa.2006.01.001.
[4]	S. Bouyuklieva, N. Yankov and R. Russeva, On the classication of binary self-dual [44, 22, 8] codes with an automorphism of order 3 or 7, Int. J. Inform. Coding Theory, 2 (2011), 21-37. doi: 10.1504/IJICOT.2011.044676.
[5]	N. Chigira, M. Harada and M. Kitazume, Extremal self-dual codes of length 64 through neighbors and covering radii, Des. Codes Crypt., 42 (2007), 93-101. doi: 10.1007/s10623-006-9018-5.
[6]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[7]	W. C. Huffman, Automorphisms of codes with application to extremal doubly-even codes of length 48, IEEE Trans. Inform. Theory, 28 (1982), 511-521. doi: 10.1109/TIT.1982.1056499.
[8]	W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490. doi: 10.1016/j.ffa.2005.05.012.
[9]	W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003.
[10]	V. I. Iorgov, Doubly even extremal codes of length 64, Probl. Inform. Transm., 22 (1986), 277-284.
[11]	S. Karadeniz and B. Yildiz, Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$, Adv. Math. Commun., 6 (2012), 193-202. doi: 10.3934/amc.2012.6.193.
[12]	G. Pasquier, A binary extremal doubly even self-dual code (64,32,12) obtained from an extended Reed-Solomon code over $F_{16}$, IEEE Trans. Inform. Theory, 27 (1981), 807-808. doi: 10.1109/TIT.1981.1056421.
[13]	R. Russeva and N. Yankov, On Binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9, Des. Codes Crypt., 45 (2007), 335-346. doi: 10.1007/s10623-007-9127-9.
[14]	N. Yankov, On binary self-dual codes of length 62 with an automorphism of order 7, Math. Educ. Math., 40 (2011), 223-228.
[15]	N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inform. Theory, 56 (2011), 7498-7506. doi: 10.1109/TIT.2011.2155619.
[16]	V. Yorgov, Binary self-dual codes with an automorphism of odd order (in Russian), Probl. Inform. Transm., 4 (1983), 13-24.

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References:

[1]	I. Bouyukliev, About the code equivalence, in Advances in Coding Theory and Cryptography, World Scientific Publishing, 2007. doi: 10.1142/9789812772022_0009.
[2]	S. Bouyuklieva, N. Yankov and J.-L. Kim, Classification of binary self-dual [48, 24, 10] codes with an automorphism of odd prime order, Finite Fields Appl., 18 (2012), 1104-1113. doi: 10.1016/j.ffa.2012.08.002.
[3]	S. Bouyuklieva, N. Yankov and R. Russeva, Classification of the binary self-dual codes having an automorphism of order 3, Finite Fields Appl., 13 (2007), 605-615. doi: 10.1016/j.ffa.2006.01.001.
[4]	S. Bouyuklieva, N. Yankov and R. Russeva, On the classication of binary self-dual [44, 22, 8] codes with an automorphism of order 3 or 7, Int. J. Inform. Coding Theory, 2 (2011), 21-37. doi: 10.1504/IJICOT.2011.044676.
[5]	N. Chigira, M. Harada and M. Kitazume, Extremal self-dual codes of length 64 through neighbors and covering radii, Des. Codes Crypt., 42 (2007), 93-101. doi: 10.1007/s10623-006-9018-5.
[6]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[7]	W. C. Huffman, Automorphisms of codes with application to extremal doubly-even codes of length 48, IEEE Trans. Inform. Theory, 28 (1982), 511-521. doi: 10.1109/TIT.1982.1056499.
[8]	W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490. doi: 10.1016/j.ffa.2005.05.012.
[9]	W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003.
[10]	V. I. Iorgov, Doubly even extremal codes of length 64, Probl. Inform. Transm., 22 (1986), 277-284.
[11]	S. Karadeniz and B. Yildiz, Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$, Adv. Math. Commun., 6 (2012), 193-202. doi: 10.3934/amc.2012.6.193.
[12]	G. Pasquier, A binary extremal doubly even self-dual code (64,32,12) obtained from an extended Reed-Solomon code over $F_{16}$, IEEE Trans. Inform. Theory, 27 (1981), 807-808. doi: 10.1109/TIT.1981.1056421.
[13]	R. Russeva and N. Yankov, On Binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9, Des. Codes Crypt., 45 (2007), 335-346. doi: 10.1007/s10623-007-9127-9.
[14]	N. Yankov, On binary self-dual codes of length 62 with an automorphism of order 7, Math. Educ. Math., 40 (2011), 223-228.
[15]	N. Yankov and R. Russeva, Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inform. Theory, 56 (2011), 7498-7506. doi: 10.1109/TIT.2011.2155619.
[16]	V. Yorgov, Binary self-dual codes with an automorphism of odd order (in Russian), Probl. Inform. Transm., 4 (1983), 13-24.

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