New binary self-dual codes of lengths 80, 84 and 96 from composite matrices
Pages 317 - 342
Abstract
In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singly-even binary self-dual codes of lengths 80, 84 and 96 as well as doubly-even binary self-dual codes of length 96 that were not known in the literature before.
References
[1]
Bortos M., Gildea J., Kaya A., Korban A., Tylyshchak A.: New self-dual codes of length 68 from a block matrix construction and group rings. Adv. Math. Commun. (2020).
[2]
Bosma W, Cannon J, and Playoust C The Magma Algebra System I: the user language J. Symb. Comput. 1997 24 3–4 235-265
[3]
Bouyukliev IG What is Q-extension? Serdica J. Comput. 2007 1 2 115-130
[4]
Dontcheva R On the doubly-even self-dual codes of length 96 IEEE Trans. Inf. Theory 2002 48 2 557-561
[5]
Dougherty S.T.: Algebraic Coding Theory over Finite Commutative Rings, 1st edn. Springer, Cham (2017).
[6]
Dougherty ST, Gulliver TA, and Harada M Extremal binary self-dual codes IEEE Trans. Inf. Theory 1997 43 6 2036-2047
[7]
Dougherty ST, Gaborit P, Harada M, and Solé P Type II codes over IEEE Trans. Inf. Theory 1999 45 1 32-45
[8]
Dougherty ST, Gildea J, and Kaya A bordered constructions of self-dual codes from group rings Finite Fields Appl. 2020
[9]
Dougherty S.T., Gildea J., Korban A., Kaya A.: Composite matrices from group rings, composite G-codes and constructions of self-dual codes (2020). arXiv:2002.11614.
[10]
Dougherty ST, Gildea J, Korban A, and Kaya A New extremal self-dual binary codes of length 68 via composite construction, lifts, extensions and neighbours Int. J. Inf. Coding Theory 2020 5 3–4 211-226
[11]
Dougherty S.T., Gildea J., Korban A.: Extending an established isomorphism between group rings and a subring of the matrices. Int. J. Algebra Comput. (2021).
[12]
Feit W A self-dual even code IEEE Trans. Inf. Theory 1974 20 1 136-138
[13]
Gaborit P, Pless V, Solé P, and Atkin O Type II codes over Finite Fields Appl. 2002 8 2 171-183
[14]
Gildea J., Kaya A., Tylyshchak A., Yildiz B.: A group induced four-circulant construction for self-dual codes and new extremal binary self-dual codes (2019). arXiv:1912.11758.
[15]
Gildea J, Kaya A, Korban A, and Tylyshchak A Self-dual codes using bisymmetric matrices and group rings Discret. Math. 2020
[16]
Gildea J, Korban A, Kaya A, and Yildiz B Constructing self-dual codes from group rings and reverse circulant matrices Adv. Math. Commun. 2020
[17]
Gildea J, Taylor R, Kaya A, and Tylyshchak A Double bordered constructions of self-dual codes from group rings over Frobenius rings Cryptogr. Commun. 2020 12 4 769-784
[18]
Gildea J, Korban A, and Roberts AM New binary self-dual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction Finite Fields Appl. 2021
[19]
Gildea J., Korban A., Roberts A.M.: Generator matrix database (2021). https://amr3-ys3da62trb.netlify.app.
[20]
Gulliver TA and Harada M Classification of extremal double circulant self-dual codes of lengths 74–88 Discret. Math. 2006 306 17 2064-2072
[21]
Gulliver TA and Harada M On extremal double circulant self-dual codes of lengths 90–96 Appl. Algebra Eng. Commun. Comput. 2019 30 5 403-415
[22]
Harada M., Yorgova R.: Construction of a self-dual code. In: Eleventh International Workshop on Algebraic and Combinatorial Coding Theory, pp. 125–128. Pamporovo, Bulgaria (2008).
[23]
Hurley T Group rings and rings of matrices Int. J. Pure Appl. Math. 2006 31 3 319-335
[24]
Kaya A and Yildiz B Various constructions for self-dual codes over rings and new binary self-dual codes Discret. Math. 2016 339 2 460-469
[25]
Kaya A, Yildiz B, and Siap I New extremal binary self-dual codes of length 68 from quadratic residue codes over Finite Fields Appl. 2014 29 160-177
[26]
Korban A., Şahinkaya S., Ustun D.: New type I Binary self-dual codes from composite matrices and Lifts (2021). arXiv:2102.00474.
[27]
Mallows CL and Sloane NJA An upper bound for self-dual codes Inf. Control 1973 22 2 188-200
[28]
Rains EM Shadow bounds for self-dual codes IEEE Trans. Inform. Theory 1998 44 1 134-139
[29]
Roberts, A.M.: Constructions of extremal and optimal self-dual and Hermitian self-dual codes over finite fields using circulant matrices. Master’s thesis, University of Chester, Chester, UK (2020). https://drive.google.com/file/d/1CMjnuBvQtrXOY8foy6_gfXOcFFuHAaFs/view.
[30]
Roberts, A.M.: Weight enumerator parameter database for binary self-dual codes (2021). https://amr-wepd-bsdc.netlify.app.
[31]
Wood JA Duality for modules over finite rings and applications to coding theory Am. J. Math. 1999 121 3 555-575
[32]
Yankov N.: Some new self-dual codes with an automorphism of order 15. Annu. Konstantin Preslavsky Univ. Shumen. XVI C:99–108 (2014).
[33]
Yankov N, Anev D, and Gürel M Self-dual codes with an automorphism of order 13 Adv. Math. Commun. 2017 11 3 635-645
Recommendations
Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes
AbstractWe introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + u F 2, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual ...
Highlights- A novel bordered construction for self-dual codes can be obtained using groups rings.
Self-dual codes from a block matrix construction characterised by group rings
AbstractWe give a new technique for constructing self-dual codes based on a block matrix whose blocks arise from group rings and orthogonal matrices. The technique can be used to construct self-dual codes over finite commutative Frobenius rings of ...
Group rings, G-codes and constructions of self-dual and formally self-dual codes
We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
© The Author(s) 2021.
Publisher
Kluwer Academic Publishers
United States
Publication History
Published: 01 February 2022
Accepted: 08 November 2021
Revision received: 19 August 2021
Received: 19 April 2021
Author Tags
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 13 Dec 2024