Quadruple bordered constructions of self-dual codes from group rings

Dougherty, Steven T.; Gildea, Joseph; Kaya, Abidin

doi:10.1007/s12095-019-00380-8

Cited by 17 publications

(30 citation statements)

References 26 publications

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“…The first examples of codes with a γ = 7 in W 68,2 are constructed in [27]. Together with these the existence of the codes in W 68,2 is known for the following parameters (see [14,8,9,26,27]): 0, 7, 11, 14, 17, 21, . .…”

Section: Computational Resultsmentioning

confidence: 99%

Constructing self-dual codes from group rings and reverse circulant matrices

Gildea¹,

Korban²,

Kaya³

et al. 2021

AMC

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In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary selfdual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

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Section: Computational Resultsmentioning

confidence: 99%

Constructing self-dual codes from group rings and reverse circulant matrices

Gildea¹,

Korban²,

Kaya³

et al. 2021

AMC

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“…The first examples of codes with γ = 8, 9 in W 68,2 are constructed in [12]. Together with these the existence of the codes in W 68,2 is known for the following parameters (see [6,5,7,20,21,11]): 7, 11, 14, 17, 20, 21, . .…”

Section: Computational Resultsmentioning

confidence: 99%

New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings

Bortos¹,

Gildea²,

Kaya³

et al. 2022

AMC

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Many generator matrices for constructing extremal binary selfdual codes of different lengths have the form G = (In | A), where In is the n×n identity matrix and A is the n×n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F 2 +uF 2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual [68, 34, 12]-codes with the rare parameters γ = 7, 8 and 9 in W 68,2. In particular, we find 92 new binary self-dual [68, 34, 12]-codes.

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“…We recall that the possible weight enumerators for a self-dual Type I [64, 32, 12]-code is given in [4,10] (2, 9) (1, E, D) (4, E) (4, F, F ) 2 3 · 3 β = 37 10 (0, A) (0, 9, 9) (4, 7) (0, 1, 5) 2 4 · 3 2 β = 40 11 (0, A) (0, 9,9) (4, 7) (2, 9, F ) 2 4 · 3 β = 64…”

Section: Constructions Coming From Cmentioning

confidence: 99%

“…(0, 0) (0, 0, 0, 0, 0, 1, 1) (0, 1) (0, 1, 1, 0, 0, 1, 1) 2 12 · 3 · 7 [32, 16,8] II (0, 0) (0, 0, 1, 0, 1, 1, 1) (0, 1) (0, 1, 1, 1, 1, 1, 1 16,8] II (1, 0) (0, 0, 0, 0, 1, 1, 1) (1, 1) (1, 1, 0, 1, 0, 1, 1 Table 4: Extremal binary self-dual codes of length 64 from self-dual codes over F 2 + uF 2 of length 32 via C 7 .…”

Section: Constructions Coming From Cmentioning

confidence: 99%

An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

Gildea

Kaya

Yıldız

2019

Discrete Mathematics

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We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F 2 + uF 2 and F 4 + uF 4 ; using groups of order 3, 7, 9, 13, and 15. Through these constructions and their extensions, we find binary self-dual codes of lengths 32, 40, * This author's visit was partially supported by the London Mathematical Society (International Short Visits -Scheme 5).† Corresponding author 56, 64, 68 and 80, all of which are extremal or optimal. In particular, we find five new self-dual codes of parameters [56, 28,10], twenty-three extremal binary self-dual codes of length 68 with new weight enumerators and fifteen new self-dual codes of parameters [80, 40,14].

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Quadruple bordered constructions of self-dual codes from group rings

Cited by 17 publications

References 26 publications

Constructing self-dual codes from group rings and reverse circulant matrices

Constructing self-dual codes from group rings and reverse circulant matrices

New self-dual codes of length 68 from a $ 2 \times 2 $ block matrix construction and group rings

An altered four circulant construction for self-dual codes from group rings and new extremal binary self-dual codes I

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