New extremal binary self-dual codes of length 68 from R_2-lifts of binary self-dual codes

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May 2013, 7(2): 219-229. doi: 10.3934/amc.2013.7.219

New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes

Suat Karadeniz ^1, and Bahattin Yildiz ^1,

1.	Department of Mathematics, Fatih University, 34500, Istanbul

Received January 2013 Published May 2013

Abstract
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A lift of binary self-dual codes to the ring $R_2$ is described. By using this lift, a family of self-dual codes over $R_2$ of length $17$ are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length $68$ are obtained. For the first time in the literature, extremal binary codes of length $68$ with $\gamma=4$ and $\gamma = 6$ in $W_{68,2}$ have been obtained. In addition to these, six new codes with $\gamma = 0$ and fourteen new codes with $\gamma = 2$ in $W_{68,2}$ have also been found.

Keywords: self-dual codes., codes over rings, Extremal codes, gray map.

Mathematics Subject Classification: Primary: 94B05; Secondary: 94B9.

Citation: Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219

References:

[1]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[2]	S. Bouyuklieva, Some Optimal self-orthogonal and self-dual codes, J. Discrete Math., 287 (2004), 1-10. doi: 10.1016/j.disc.2004.06.010.
[3]	S. Bouyuklieva and I. Boukliev, 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, N. Yankov and J.-L. Kim, Classification of binary self-dual [48,24,10]-codes with an automorphism odd prime order, Finite Fields Appl., 18 (2012), 1104-1113. doi: 10.1016/j.ffa.2012.08.002.
[5]	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.
[6]	S. T. Dougherty, A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Infrom. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.
[7]	S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 4-26. doi: 10.1016/j.ffa.2009.11.003.
[8]	S. T. Dougherty, J.-L. Kim and H. Liu, Constructions of self-dual codes over finite commutative chain rings, J. Inform. Coding Theory, 1 (2010), 171-190. doi: 10.1504/IJICOT.2010.032133.
[9]	S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219. doi: 10.1016/j.ffa.2010.11.002.
[10]	S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, European J. Pure Appl. Math., 6 (2013), 89-106.
[11]	P. Gaborit and A. Otmani, Experimental constructions of self-dual codes, Finite Fields Appl., 9 (2003), 372-394. doi: 10.1016/S1071-5797(03)00011-X.
[12]	S. Karadeniz and B. Yildiz, $R_2$-generator matrices for extremal self-dual codes of length 68,, available online at \url{http://www.fatih.edu.tr/~akaya/NewSelf-dual68.pdf}, ().
[13]	S. Karadeniz and B. Yildiz, Double-circulant and double-bordered-circulant constructions for self-dual codes over $R_2$, Adv. Math. Commun., 6 (2012), 193-202. doi: 10.3934/amc.2012.6.193.
[14]	S. Karadeniz and B. Yildiz, New extremal binary self-dual codes of length $64$ as $R_3$-lifts of the extended binary Hamming code,, submitted., ().
[15]	H. H. Kim, H. Lee, J. B. Lee and Y. Lee, Construction of self-dual codes with an automorpihsm of of order $p$, Adv. Math. Commun., 5 (2011), 23-36. doi: 10.3934/amc.2011.5.23.
[16]	A. Munemasa, Database of self-dual codes,, available online at \url{http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/data/2/18.magma}, ().
[17]	T. Nishimura, A new extremal self-dual code of length 64, IEEE Trans. Inform. Theory, 50 (2004), 2173-2174. doi: 10.1109/TIT.2004.833359.
[18]	E. M. Rains, Shadow bounds for self dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[19]	H. P. Tsai, P. Y. Shih, R. Y. Wuh, W. K. Su and C. H. Chen, Construction of self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 3826-3831. doi: 10.1109/TIT.2008.926454.
[20]	J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575. doi: 10.1353/ajm.1999.0024.
[21]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, Des. Codes Crypt., 54 (2010), 61-81. doi: 10.1007/s10623-009-9309-8.
[22]	B. Yildiz and S. Karadeniz, Self-dual codes over $\mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, J. Franklin Inst., 347 (2010), 1888-1894. doi: 10.1016/j.jfranklin.2010.10.007.

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References:

[1]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[2]	S. Bouyuklieva, Some Optimal self-orthogonal and self-dual codes, J. Discrete Math., 287 (2004), 1-10. doi: 10.1016/j.disc.2004.06.010.
[3]	S. Bouyuklieva and I. Boukliev, 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, N. Yankov and J.-L. Kim, Classification of binary self-dual [48,24,10]-codes with an automorphism odd prime order, Finite Fields Appl., 18 (2012), 1104-1113. doi: 10.1016/j.ffa.2012.08.002.
[5]	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.
[6]	S. T. Dougherty, A. Gulliver and M. Harada, Extremal binary self-dual codes, IEEE Trans. Infrom. Theory, 43 (1997), 2036-2047. doi: 10.1109/18.641574.
[7]	S. T. Dougherty, J.-L. Kim, H. Kulosman and H. Liu, Self-dual codes over commutative Frobenius rings, Finite Fields Appl., 16 (2010), 4-26. doi: 10.1016/j.ffa.2009.11.003.
[8]	S. T. Dougherty, J.-L. Kim and H. Liu, Constructions of self-dual codes over finite commutative chain rings, J. Inform. Coding Theory, 1 (2010), 171-190. doi: 10.1504/IJICOT.2010.032133.
[9]	S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219. doi: 10.1016/j.ffa.2010.11.002.
[10]	S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, European J. Pure Appl. Math., 6 (2013), 89-106.
[11]	P. Gaborit and A. Otmani, Experimental constructions of self-dual codes, Finite Fields Appl., 9 (2003), 372-394. doi: 10.1016/S1071-5797(03)00011-X.
[12]	S. Karadeniz and B. Yildiz, $R_2$-generator matrices for extremal self-dual codes of length 68,, available online at \url{http://www.fatih.edu.tr/~akaya/NewSelf-dual68.pdf}, ().
[13]	S. Karadeniz and B. Yildiz, Double-circulant and double-bordered-circulant constructions for self-dual codes over $R_2$, Adv. Math. Commun., 6 (2012), 193-202. doi: 10.3934/amc.2012.6.193.
[14]	S. Karadeniz and B. Yildiz, New extremal binary self-dual codes of length $64$ as $R_3$-lifts of the extended binary Hamming code,, submitted., ().
[15]	H. H. Kim, H. Lee, J. B. Lee and Y. Lee, Construction of self-dual codes with an automorpihsm of of order $p$, Adv. Math. Commun., 5 (2011), 23-36. doi: 10.3934/amc.2011.5.23.
[16]	A. Munemasa, Database of self-dual codes,, available online at \url{http://www.math.is.tohoku.ac.jp/~munemasa/research/codes/data/2/18.magma}, ().
[17]	T. Nishimura, A new extremal self-dual code of length 64, IEEE Trans. Inform. Theory, 50 (2004), 2173-2174. doi: 10.1109/TIT.2004.833359.
[18]	E. M. Rains, Shadow bounds for self dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[19]	H. P. Tsai, P. Y. Shih, R. Y. Wuh, W. K. Su and C. H. Chen, Construction of self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 3826-3831. doi: 10.1109/TIT.2008.926454.
[20]	J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575. doi: 10.1353/ajm.1999.0024.
[21]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, Des. Codes Crypt., 54 (2010), 61-81. doi: 10.1007/s10623-009-9309-8.
[22]	B. Yildiz and S. Karadeniz, Self-dual codes over $\mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, J. Franklin Inst., 347 (2010), 1888-1894. doi: 10.1016/j.jfranklin.2010.10.007.

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