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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

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

[1]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[2]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018

[3]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[4]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[5]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[6]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127

[7]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[8]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[9]

Xi Zhao, Teng Niu. Impacts of horizontal mergers on dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020173

[10]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285

[11]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309

[12]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381

[13]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[14]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[15]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[16]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001

[17]

Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229

[18]

Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167

[19]

Hongxia Sun, Yao Wan, Yu Li, Linlin Zhang, Zhen Zhou. Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours. Journal of Industrial & Management Optimization, 2021, 17 (2) : 601-631. doi: 10.3934/jimo.2019125

[20]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]