May  2012, 6(2): 193-202. doi: 10.3934/amc.2012.6.193

Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$

1. 

Department of Mathematics, Fatih University, 34500, Istanbul, Turkey, Turkey

Received  April 2011 Revised  July 2011 Published  April 2012

In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructions for self-dual codes over the non-chain ring $R_2 = \mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$ are introduced. Using these methods, we have constructed three extremal binary Type I codes of length $64$ of new weight enumerators for which extremal codes were not known to exist. We also give a double-circulant construction for extremal binary self-dual codes of length $40$ with covering radius $7$.
Citation: Suat Karadeniz, Bahattin Yildiz. Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$. Advances in Mathematics of Communications, 2012, 6 (2) : 193-202. doi: 10.3934/amc.2012.6.193
References:
[1]

S. Bouyuklieva, Some Optimal self-orthogonal and self-dual codes,, J. Discrete Math., 287 (2004), 1. doi: 10.1016/j.disc.2004.06.010.

[2]

S. Bouyuklieva and V. Yorgov, Singly-even self-dual codes of length $40$,, Des. Codes Cryptogr., 9 (1996), 131. doi: 10.1007/BF00124589.

[3]

N. Chigira, M. Harada and M. Kitazume, Extremal self-dual codes of length 64 through neighbors and covering radii,, Des. Codes Cryptogr., 42 (2007), 93. doi: 10.1007/s10623-006-9018-5.

[4]

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. doi: 10.1109/18.59931.

[5]

D. B. Dalan, New Extremal Type I Codes of lengths 40, 42 and 44,, Des. Codes Cryptogr., 30 (2003), 151. doi: 10.1023/A:1025476619824.

[6]

S. T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type II codes over $\mathbb F_2+u\mathbb F_2$,, IEEE Trans. Infrom. Theory, 45 (1999), 32.

[7]

P. Gaborit and A. Otmani, Experimental constructions of self-dual codes,, Finite Fields Appl., 9 (2003), 372.

[8]

T. A. Gulliver, Construction of optimal Type IV self-dual codes over $\mathbb F_2+u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 2520. doi: 10.1109/18.796394.

[9]

M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to 62,, J. Discrete Math., 188 (1998), 127. doi: 10.1016/S0012-365X(97)00250-1.

[10]

M. Harada, M. Kiermaier, A. Wasserman and R. Yorgova, New binary singly even self-dual codes,, IEEE Trans. Inform. Theory, 56 (2010), 1612. doi: 10.1109/TIT.2010.2040967.

[11]

M. Harada, A. Munemasa and K. Tanabe, Extremal self-dual [40,20,8] codes with covering radius 7,, Finite Fields Appl., 10 (2004), 183. doi: 10.1016/j.ffa.2003.08.001.

[12]

M. Harada and M. Ozeki, Extremal self-dual codes with the smallest covering radius,, Discrete Math., 215 (2000), 271. doi: 10.1016/S0012-365X(99)00318-0.

[13]

T. Nishimura, A new extremal self-dual code of length 64,, IEEE Trans. Inform. Theory, 50 (2004), 2173. doi: 10.1109/TIT.2004.833359.

[14]

M. Ozeki, On covering radii and coset weight distributions of extremal binary self-dual codes of length 40,, Theoret. Comput. Sci., 235 (2000), 283. doi: 10.1016/S0304-3975(99)00200-5.

[15]

H. P. Tsai, P. Y. Shih, R. Y. Wuh, W. K. Su and C. H. Chen, Construction of self-fual codes,, IEEE Trans. Inform. Theory, 54 (2008), 3826. doi: 10.1109/TIT.2008.926454.

[16]

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 Cryptogr., 54 (2010), 61. doi: 10.1007/s10623-009-9309-8.

[17]

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. doi: 10.1016/j.jfranklin.2010.10.007.

[18]

V. I. Yorgo and N. Ziapkov, Doubly even self-dual [40,20,8] codes with automorphism of an odd order,, Probl. Peredachi Inf., 32 (1996), 41.

[19]

R. Yorgova, Constructing self-dual codes using an automorphism group,, in, (2006).

show all references

References:
[1]

S. Bouyuklieva, Some Optimal self-orthogonal and self-dual codes,, J. Discrete Math., 287 (2004), 1. doi: 10.1016/j.disc.2004.06.010.

[2]

S. Bouyuklieva and V. Yorgov, Singly-even self-dual codes of length $40$,, Des. Codes Cryptogr., 9 (1996), 131. doi: 10.1007/BF00124589.

[3]

N. Chigira, M. Harada and M. Kitazume, Extremal self-dual codes of length 64 through neighbors and covering radii,, Des. Codes Cryptogr., 42 (2007), 93. doi: 10.1007/s10623-006-9018-5.

[4]

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. doi: 10.1109/18.59931.

[5]

D. B. Dalan, New Extremal Type I Codes of lengths 40, 42 and 44,, Des. Codes Cryptogr., 30 (2003), 151. doi: 10.1023/A:1025476619824.

[6]

S. T. Dougherty, P. Gaborit, M. Harada and P. Solé, Type II codes over $\mathbb F_2+u\mathbb F_2$,, IEEE Trans. Infrom. Theory, 45 (1999), 32.

[7]

P. Gaborit and A. Otmani, Experimental constructions of self-dual codes,, Finite Fields Appl., 9 (2003), 372.

[8]

T. A. Gulliver, Construction of optimal Type IV self-dual codes over $\mathbb F_2+u\mathbb F_2$,, IEEE Trans. Inform. Theory, 45 (1999), 2520. doi: 10.1109/18.796394.

[9]

M. Harada, T. A. Gulliver and H. Kaneta, Classification of extremal double-circulant self-dual codes of length up to 62,, J. Discrete Math., 188 (1998), 127. doi: 10.1016/S0012-365X(97)00250-1.

[10]

M. Harada, M. Kiermaier, A. Wasserman and R. Yorgova, New binary singly even self-dual codes,, IEEE Trans. Inform. Theory, 56 (2010), 1612. doi: 10.1109/TIT.2010.2040967.

[11]

M. Harada, A. Munemasa and K. Tanabe, Extremal self-dual [40,20,8] codes with covering radius 7,, Finite Fields Appl., 10 (2004), 183. doi: 10.1016/j.ffa.2003.08.001.

[12]

M. Harada and M. Ozeki, Extremal self-dual codes with the smallest covering radius,, Discrete Math., 215 (2000), 271. doi: 10.1016/S0012-365X(99)00318-0.

[13]

T. Nishimura, A new extremal self-dual code of length 64,, IEEE Trans. Inform. Theory, 50 (2004), 2173. doi: 10.1109/TIT.2004.833359.

[14]

M. Ozeki, On covering radii and coset weight distributions of extremal binary self-dual codes of length 40,, Theoret. Comput. Sci., 235 (2000), 283. doi: 10.1016/S0304-3975(99)00200-5.

[15]

H. P. Tsai, P. Y. Shih, R. Y. Wuh, W. K. Su and C. H. Chen, Construction of self-fual codes,, IEEE Trans. Inform. Theory, 54 (2008), 3826. doi: 10.1109/TIT.2008.926454.

[16]

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 Cryptogr., 54 (2010), 61. doi: 10.1007/s10623-009-9309-8.

[17]

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. doi: 10.1016/j.jfranklin.2010.10.007.

[18]

V. I. Yorgo and N. Ziapkov, Doubly even self-dual [40,20,8] codes with automorphism of an odd order,, Probl. Peredachi Inf., 32 (1996), 41.

[19]

R. Yorgova, Constructing self-dual codes using an automorphism group,, in, (2006).

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