Some results on free Euclidean self-dual codes over F2+vF2

Year 2019, , 1131 - 1136, 01.12.2019

Refia Aksoy Fatma Çalışkan

https://doi.org/10.16984/saufenbilder.525606

Abstract

 In this
paper, free Euclidean self-dual codes over the ring  F₂ + v F₂  with v² =v of order 4 are considered. A necessary and
sufficient condition for the form of the generator matrix of a free Euclidean
self-dual code is given. By using the distance preserving Gray map from  F₂ + v F₂  to F₂ x  F₂,
the generator matrix of the binary code which corresponds the code over the
ring F₂ + v F₂  is obtained. The codes of lengths up to 100 over
the ring F₂ + v F₂ are
found.

Keywords

Euclidean self-dual codes, codes over rings, Chinese remainder theorem

References

• C. Bachoc, “Applications of coding theory to the construction of modular lattices,” J Combin Theory Ser A, vol. 78, pp. 92-119, 1997.
• K. Betsumiya and M. Harada, “Optimal self-dual codes over F_2×F_2 with respect to the Hamming weight,” IEEE Trans Inform Theory, vol. 50, pp. 356-358, 2004.
• W. Bosma, J. Cannon and C. Playoust, “The Magma algebra system I: The user language,” J Symbolic Comput, vol. 24, pp. 235-265, 1997.
• A. E. Brouwer, Bounds on the size of linear codes, in: V. S. Pless, W. C. Huffman (Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, pp. 295-461, 1998.
• Y. Cengellenmis, A. Dertli and S. T. Dougherty, “Codes over an infinite family of rings with a Gray map,” Des Codes Cryptogr, vol. 72, pp. 559-580, 2014.
• J. H. Conway, N. J. A. Sloane, “A New Upper Bound on Minimal Distance of Self-Dual Codes,” IEEE Trans Inform Theory, vol. 36, pp. 1319-1333, 1990.
• S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa and P. Sole ́, “Type IV self-dual codes over rings,” IEEE Trans Inform Theory, vol. 45, pp. 2345-2360, 1999.
• S. T. Dougherty, J. L. Kim, H. Liu, “Constructions of self-dual codes over finite commutative chain rings,” Int. J. Inf. Coding Theory 1, vol. 2, pp. 171-190, 2010.
• S. T. Dougherty, M. Harada and P. Sole ́, “Self-dual codes over rings and the Chinese remainder theorem,” Hokkaido Math. J., vol. 28, pp. 253-283, 1999.
• J. Gao, Y. Wang, J. Li, “Bounds on covering radius of linear codes with Chinese Euclidean distance over the finite non chain ring F_2+υF_2,” Inf. Process. Lett., vol. 138, pp. 22-26, 2018.
• J. Gildea, A. Kaya, R. Taylor, B. Yildiz, “Constructions for self-dual codes induced from group rings,” Finite Fields Appl., vol. 51, pp. 71—92, 2018.
• C. A. Castillo-Guille ́n, C. Renteri ́a-Ma ́rquez, H. Tapia-Recillas, “Duals of constacyclic codes over finite local Frobenius non-chain rings of length 4,” Discrete Math., vol. 341, pp. 919-933, 2018.
• R. Hill, A First Course in Coding Theory, Oxford University Press, 1986.
• S. Karadeniz, S. T. Dougherty and B. Yildiz, “Constructing formally self-dual codes over R_k,” Discrete Appl. Math., vol. 167, pp. 188-196, 2014.
• B. Kim, Y. Lee, “Lee Weights of Cyclic Self-Dual Codes over Galois Rings of Characteristic p^2,” Finite Fields Appl., vol. 45, pp. 107-130, 2017.
• J. L. Kim, Y. Lee, “An Efficient Construction of Self-Dual Codes,” Bull. Korean Math. Soc., vol. 52, pp. 915-923, 2015.
• J. Li, A. Zhang, K. Feng, “Linear Codes over (F_q [x] )⁄((x^2 ) ) and GR(p^2,m) Reaching the Griesmer Bound,” Des Codes Cryptogr, vol. 86, pp. 2837-2855, 2018.
• S. Zhu, Y. Wang and M. Shi, “Some results on cyclic codes over F_2+υF_2,” IEEE Trans Inform Theory, vol. 56, pp. 1680-1684, 2010.