New extremal singly even self-dual codes of lengths 64 and 66

Year 2018, , 143 - 151, 08.10.2018

Damyan Anev Masaaki Harada Nikolay Yankov

https://doi.org/10.13069/jacodesmath.458601

Cited By: 1

Abstract

For lengths $64$ and $66$,
we construct six and seven extremal singly even self-dual
codes with weight enumerators for which no extremal
singly even self-dual codes were previously known to exist, respectively.
We also construct new $40$ inequivalent
extremal doubly even self-dual $[64,32,12]$ codes
with covering radius $12$ meeting the Delsarte bound.
These new codes are constructed by considering
four-circulant codes along with their neighbors and shadows.

Keywords

Self-dual code, Weight enumerator

References

• [1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symb. Comput. 24(3–4) (1997) 235–265.
• [2] I. Bouyukliev, "About the code equivalence" in Advances in Coding Theory and Cryptology, NJ, Hackensack: World Scientific, 2007.
• [3] R. A. Brualdi, V. S. Pless, Weight enumerators of self–dual codes, IEEE Trans. Inform. Theory 37(4) (1991) 1222–1225.
• [4] N. Chigira, M. Harada, M. Kitazume, Extremal self-dual codes of length 64 through neighbors and covering radii, Des. Codes Cryptogr. 42(1) (2007) 93–101.
• [5] P. Çomak, J. L. Kim, F. Özbudak, New cubic self–dual codes of length 54; 60 and 66, Appl. Algebra Engrg. Comm. Comput. 29(4) (2018) 303–312.
• [6] J. H. Conway, N. J. A. Sloane, A new upper bound on the minimal distance of self–dual codes, IEEE Trans. Inform. Theory 36(6) (1990) 1319–1333.
• [7] S. T. Dougherty, T. A. Gulliver, M. Harada, Extremal binary self–dual codes, IEEE Trans. Inform. Theory 43(6) (1997) 2036–2047.
• [8] M. Harada, T. Nishimura, R. Yorgova, New extremal self–dual codes of length 66, Math. Balkanica (N.S.) 21(1–2) (2007) 113–121.
• [9] S. Karadeniz, B. Yildiz, New extremal binary self–dual codes of length 66 as extensions of self–dual codes over $R_k$, J. Franklin Inst. 350(8) (2013) 1963–1973.
• [10] A. Kaya, New extremal binary self–dual codes of lengths 64 and 66 from $R_2$–lifts, Finite Fields Appl. 46 (2017) 271–279.
• [11] A. Kaya, B. Yildiz, A. Pasa, New extremal binary self–dual codes from a modified four circulant construction, Discrete Math. 339(3) (2016) 1086–1094.
• [12] A. Kaya, B. Yildiz, I. Siap, New extremal binary self–dual codes from F4 + uF4–lifts of quadratic circulant codes over F4, Finite Fields Appl. 35 (2015) 318–329.
• [13] E. M. Rains, Shadow bounds for self–dual codes, IEEE Trans. Inform. Theory 44(1) (1998) 134–139.
• [14] H.-P. Tsai, Existence of certain extremal self–dual codes, IEEE Trans. Inform. Theory 38(2) (1992) 501–504.
• [15] H.-P. Tsai, Extremal self–dual codes of lengths 66 and 68, IEEE Trans. Inform. Theory 45(6) (1999) 2129–2133.
• [16] N. Yankov, Self–dual [62; 31; 12] and [64; 32; 12] codes with an automorphism of order 7, Adv. Math. Commun. 8(1) (2014) 73–81.
• [17] N. Yankov, M. H. Lee, M. Gürel, M. Ivanova, Self–dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory 61(3) (2015) 1188–1193.
• [18] N. Yankov, M. Ivanova, M. H. Lee, Self–dual codes with an automorphism of order 7 and s–extremal codes of length 68, Finite Fields Appl. 51 (2018) 17–30.