A group ring construction of the [48,24,12] type II linear block code

McLoughlin, Ian

doi:10.1007/s10623-011-9530-0

Cited by 18 publications

(9 citation statements)

References 9 publications

(19 reference statements)

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“…For example, McLoughlin [1] provided a construction of the self-dual, doubly-even and extremal [48,24,12] binary linear block code using a zero divisor in the dihedral group algebra F 2 [D 48 ]. Dougherty et al [2] and [3] gave constructions of self-dual and formally self-dual codes from group rings R [G] where the ring R is a finite commutative Frobenius ring.…”

Section: Introductionmentioning

confidence: 99%

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Self-Dual Binary $[8m,\,\,4m]$ -Codes Constructed by Left Ideals of the Dihedral Group Algebra $\mathbb{F}_2[D_{8m}]$

Chen

Cao

et al. 2020

IEEE Trans. Inform. Theory

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Let m be an arbitrary positive integer and D8m be a dihedral group of order 8m, i.e., D8m = x, y | x 4m = 1, y 2 = 1, yxy = x −1 . Left ideals of the dihedral group algebra F2[D8m] are called binary left dihedral codes of length 8m, and abbreviated as binary left D8m-codes. In this paper, we give an explicit representation and enumeration for all distinct self-dual binary left D8m-codes. These codes make up an important class of self-dual binary [8m, 4m]-codes such that the dihedral group D8m is necessary a subgroup of the automorphism group of each code. In particular, we provide recursive algorithms to solve congruence equations over finite chain rings for constructing all distinct self-dual binary left D8m-codes and obtain a Mass formula to count the number of all these self-dual codes. As a preliminary application, we obtain the extremal self-dual binary [48, 24,12]-code and an extremal self-dual binary [56, 28, 12]code from self-dual binary left D48-codes and left D56-codes respectively.

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Section: Introductionmentioning

confidence: 99%

“…In this paper, we provide a new way to construct binary self-dual [8m, 4m]-codes which is different from the methods used in [1], [2], [3] and [4]. Specifically, we give an explicit construction and enumeration for all distinct self-dual binary left D 8m -codes.…”

Section: Introductionmentioning

confidence: 99%

Self-Dual Binary $[8m,\,\,4m]$ -Codes Constructed by Left Ideals of the Dihedral Group Algebra $\mathbb{F}_2[D_{8m}]$

Chen

Cao

et al. 2020

IEEE Trans. Inform. Theory

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“…The matrix, A, has been used in numerous construction methods to describe a linear code, [28]. This theory was well established with the realization of the [48, 24,12] extended QR code as a group ring code for the dihedral group, [27]. Notably, in 1990 [1], the extended Golay codes were constructed from ideals in group rings.…”

Section: Introductionmentioning

confidence: 99%

Double bordered constructions of self-dual codes from group rings over Frobenius rings

et al. 2020

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In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F 2 + uF 2 and F 4 + uF 4. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables. Keywords Group rings • Self-dual codes • Codes over rings • Extremal codes • Bordered constructions Mathematics Subject Classification (2010) 94B05 • 94B15 This research was supported by the London Mathematical Society (International Short Visits-Scheme 5).

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“…Group rings have also been used in constructing extremal binary self-dual codes. In [1,19] and [20], the extended binary Golay code and the extended Quadratic residue code were constructed from group rings using the symmetric group of degree four and dihedral groups. There have been more recent works in which the construction is generalized and modified to include many more groups and self-dual codes of different lengths.…”

Section: Introductionmentioning

confidence: 99%

Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes

Dougherty

Gildea

Korban

et al. 2019

Finite Fields and Their Applications

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We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + uF 2 , using groups of * corresponding author orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.

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A group ring construction of the [48,24,12] type II linear block code

Cited by 18 publications

References 9 publications

Self-Dual Binary $[8m,\,\,4m]$ -Codes Constructed by Left Ideals of the Dihedral Group Algebra $\mathbb{F}_2[D_{8m}]$

Self-Dual Binary $[8m,\,\,4m]$ -Codes Constructed by Left Ideals of the Dihedral Group Algebra $\mathbb{F}_2[D_{8m}]$

Double bordered constructions of self-dual codes from group rings over Frobenius rings

Bordered constructions of self-dual codes from group rings and new extremal binary self-dual codes

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