Construction of self-dual codes with an automorphism of order p

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February 2011, 5(1): 23-36. doi: 10.3934/amc.2011.5.23

Construction of self-dual codes with an automorphism of order $p$

Hyun Jin Kim ^1, , Heisook Lee ^2, , June Bok Lee ^1, and Yoonjin Lee ^2,

1.	Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-Gu, Seoul 120-749, South Korea, South Korea
2.	Department of Mathematics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul, 120-750, South Korea, South Korea

Received April 2010 Revised December 2010 Published February 2011

Abstract
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We develop a construction method for finding self-dual codes with an automorphism of order $p$ with $c$ independent $p$-cycles. In more detail, we construct a self-dual code with an automorphism of type $p-(c,f+2)$ and length $n+2$ from a self-dual code with an automorphism of type $p-(c,f)$ and length $n$, where an automorphism of type $p-(c, f)$ is that of order $p$ with $c$ independent cycles and $f$ fixed points. Using this construction, we find three new inequivalent extremal self-dual $[54, 27, 10]$ codes with an automorphism of type $7-(7,5)$ and two new inequivalent extremal self-dual $[58, 29, 10]$ codes with an automorphism of of type $7-(8,2)$. We also obtain an extremal self-dual $[40, 20, 8]$ code with an automorphism of type $3-(10, 10)$, which is constructed from an extremal self-dual $[38, 19, 8]$ code of type $3-(10,8)$, and at least 482 inequivalent extremal self-dual $[58,29,10]$ codes with an automorphism of type $3-(18,4)$, which is constructed from an extremal self-dual $[54, 27, 10]$ code of type $3-(18,0);$ we note that the extremality is preserved.

Keywords: extremal code, Automorphism, self-dual code, weight enumerator..

Mathematics Subject Classification: Primary: 94B05; Secondary: 11T7.

Citation: 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

References:

[1]	I. Bouyukliev and S. Bouyuklieva, Some new extremal self-dual codes with lengths $44, 50, 54,$ and $58$, IEEE Trans. Inform. Theory, 44 (1998), 809-812. doi: 10.1109/18.661526.
[2]	S. Bouyuklieva, A method for constructing self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 46 (2000), 496-504. doi: 10.1109/18.825812.
[3]	S. Bouyuklieva and I. Bouyukliev, Extremal self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 44 (1998), 323-328. doi: 10.1109/18.651059.
[4]	S. Bouyuklieva and P. Östergård, New constructions of optimal self-dual binary codes of length $54$, Des. Codes Crypt., 41 (2006), 101-109. doi: 10.1007/s10623-006-0018-2.
[5]	S. Bouyuklieva, R. Russeva and N. Yankov, On the structure of binary self-dual codes having an automorphism of order a square of an odd prime, IEEE Trans. Inform. Theory, 51 (2005), 3678-3686. doi: 10.1109/TIT.2005.855616.
[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 (1991), 1319-1333. doi: 10.1109/18.59931.
[7]	R. Dontcheva and M. Harada, Extremal self-dual codes of length 62 and related extremal self-dual codes, IEEE Trans. Inform. Theory, 48 (2002), 2060-2064. doi: 10.1109/TIT.2002.1013144.
[8]	R. Dontcheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7, Algebra Eng. Commun. Comput. (AAECC J.), 14 (2003), 75-79.
[9]	S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.
[10]	T. A. Gulliver, J.-L. Kim and Y. Lee, New MDS and near-MDS self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 4354-4360. doi: 10.1109/TIT.2008.928297.
[11]	M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to 62, Discrete Math., 188 (1998), 127-136. doi: 10.1016/S0012-365X(97)00250-1.
[12]	M. Harada and H. Kimura, On extremal self-dual codes, Math. J. Okayama Univ., 37 (1995), 1-14.
[13]	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.
[14]	W. C. Huffman, The $[52,26,10]$ binary self-dual codes with an automorphism of order $7$, Finite Fields Appl., 7 (2001), 341-349. doi: 10.1006/ffta.2000.0295.
[15]	J.-L. Kim, New extremal self-dual codes of length $36$, $38$, and $58$, IEEE Trans. Inform. Theory, 47 (2001), 386-393. doi: 10.1109/18.904540.
[16]	J.-L. Kim and Y. Lee, Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combin. Theory Ser. A, 105 (2004), 79-95. doi: 10.1016/j.jcta.2003.10.003.
[17]	V. Pless, A classification of self-orthogonal codes over $GF(2)$, Discrete Math., 3 (1972), 209-246. doi: 10.1016/0012-365X(72)90034-9.
[18]	V. Pless, N. J. A. Sloane and H. N. Ward, Ternary codes of minimum weight 6 and the classification of the self-dual codes of length 20, IEEE Trans. Inform. Theory, 26 (1980), 306-316. doi: 10.1109/TIT.1980.1056195.
[19]	H.-P. Tsai, Existence of certain extremal self-dual codes, IEEE Trans. Inform. Theory, 38 (1992), 501-504. doi: 10.1109/18.119711.
[20]	H.-P. Tsai and Y. J. Jiang, Some new extremal self-dual $[58,29,10]$ codes, IEEE Trans. Inform. Theory, 44 (1998), 813-814. doi: 10.1109/18.661527.
[21]	V. Y. Yorgov, Binary self-dual codes with automorphisms of an odd order, Problems Inform. Trans., 19 (1983), 260-270.
[22]	V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inform. Theory, 33 (1987), 77-82. doi: 10.1109/TIT.1987.1057273.
[23]	S. Zhang and S. Li, Some new extremal self-dual codes with lengths $42, 44, 52,$ and $58$, Discrete Math., 238 (2001), 147-150. doi: 10.1016/S0012-365X(00)00420-9.

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References:

[1]	I. Bouyukliev and S. Bouyuklieva, Some new extremal self-dual codes with lengths $44, 50, 54,$ and $58$, IEEE Trans. Inform. Theory, 44 (1998), 809-812. doi: 10.1109/18.661526.
[2]	S. Bouyuklieva, A method for constructing self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 46 (2000), 496-504. doi: 10.1109/18.825812.
[3]	S. Bouyuklieva and I. Bouyukliev, Extremal self-dual codes with an automorphism of order 2, IEEE Trans. Inform. Theory, 44 (1998), 323-328. doi: 10.1109/18.651059.
[4]	S. Bouyuklieva and P. Östergård, New constructions of optimal self-dual binary codes of length $54$, Des. Codes Crypt., 41 (2006), 101-109. doi: 10.1007/s10623-006-0018-2.
[5]	S. Bouyuklieva, R. Russeva and N. Yankov, On the structure of binary self-dual codes having an automorphism of order a square of an odd prime, IEEE Trans. Inform. Theory, 51 (2005), 3678-3686. doi: 10.1109/TIT.2005.855616.
[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 (1991), 1319-1333. doi: 10.1109/18.59931.
[7]	R. Dontcheva and M. Harada, Extremal self-dual codes of length 62 and related extremal self-dual codes, IEEE Trans. Inform. Theory, 48 (2002), 2060-2064. doi: 10.1109/TIT.2002.1013144.
[8]	R. Dontcheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7, Algebra Eng. Commun. Comput. (AAECC J.), 14 (2003), 75-79.
[9]	S. T. Dougherty, T. A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.
[10]	T. A. Gulliver, J.-L. Kim and Y. Lee, New MDS and near-MDS self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 4354-4360. doi: 10.1109/TIT.2008.928297.
[11]	M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to 62, Discrete Math., 188 (1998), 127-136. doi: 10.1016/S0012-365X(97)00250-1.
[12]	M. Harada and H. Kimura, On extremal self-dual codes, Math. J. Okayama Univ., 37 (1995), 1-14.
[13]	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.
[14]	W. C. Huffman, The $[52,26,10]$ binary self-dual codes with an automorphism of order $7$, Finite Fields Appl., 7 (2001), 341-349. doi: 10.1006/ffta.2000.0295.
[15]	J.-L. Kim, New extremal self-dual codes of length $36$, $38$, and $58$, IEEE Trans. Inform. Theory, 47 (2001), 386-393. doi: 10.1109/18.904540.
[16]	J.-L. Kim and Y. Lee, Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combin. Theory Ser. A, 105 (2004), 79-95. doi: 10.1016/j.jcta.2003.10.003.
[17]	V. Pless, A classification of self-orthogonal codes over $GF(2)$, Discrete Math., 3 (1972), 209-246. doi: 10.1016/0012-365X(72)90034-9.
[18]	V. Pless, N. J. A. Sloane and H. N. Ward, Ternary codes of minimum weight 6 and the classification of the self-dual codes of length 20, IEEE Trans. Inform. Theory, 26 (1980), 306-316. doi: 10.1109/TIT.1980.1056195.
[19]	H.-P. Tsai, Existence of certain extremal self-dual codes, IEEE Trans. Inform. Theory, 38 (1992), 501-504. doi: 10.1109/18.119711.
[20]	H.-P. Tsai and Y. J. Jiang, Some new extremal self-dual $[58,29,10]$ codes, IEEE Trans. Inform. Theory, 44 (1998), 813-814. doi: 10.1109/18.661527.
[21]	V. Y. Yorgov, Binary self-dual codes with automorphisms of an odd order, Problems Inform. Trans., 19 (1983), 260-270.
[22]	V. Y. Yorgov, A method for constructing inequivalent self-dual codes with applications to length 56, IEEE Trans. Inform. Theory, 33 (1987), 77-82. doi: 10.1109/TIT.1987.1057273.
[23]	S. Zhang and S. Li, Some new extremal self-dual codes with lengths $42, 44, 52,$ and $58$, Discrete Math., 238 (2001), 147-150. doi: 10.1016/S0012-365X(00)00420-9.

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