Construction for both self-dual codes and LCD codes

Ishizuka, Keita; Saito, Ken

doi:10.48550/arxiv.2108.12544

Cited by 3 publications

(9 citation statements)

References 26 publications

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“…In this paper, we also give a new method for constructing many Type II Z 2k -codes starting from a given Type II Z 2k -code with generator matrix I n A by transforming A. This generalizes the methods given in [14] and [23]. By four-negacirculant codes and by the new method, we construct extremal Type II Z 2k -codes of length 24 for k = 4, 5, .…”

Section: Introductionmentioning

confidence: 92%

“…Remark 3.5. For the case k = 1, the above theorem can be found in [23]. Theorem 5 in [14] can be obtained from the above theorem by putting k = 2, x = t and y = (2, 2, .…”

Section: Preliminariesmentioning

confidence: 99%

“…For example, starting from a given self-dual code with generator matrix I n A , by transforming A, some methods for constructing many self-dual codes are known, where I n denotes the identity matrix of order n (see e.g. [14], [19], [23] and [27]). In this paper, we give a new method for constructing many self-dual Z m -codes starting from a given self-dual Z m -code with generator matrix I n A by transforming A.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we give a new method for constructing many self-dual Z m -codes starting from a given self-dual Z m -code with generator matrix I n A by transforming A. For self-dual codes over finite fields, the corresponding result can be found in [23]. Now let us consider the case m = 2k.…”

Section: Introductionmentioning

confidence: 99%

“…. , 20) M 34,24,2 (32,25,16,3,16,22,27, 29, 0, 0, 12, 18) (31,16,17,3,25,17, 29,8, 30,9, 32,13) C 34,24,8 M 34,24,4(15, 20, 3, 23, 6, 23, 6, 6, 0, 12, 20, 28) (6, 26, 25, 22, 4, 7, 28, 6, 15, 14, 30, 23) C 34,24,9 M 34,24,4 (13, 10, 26, 7, 11, 19, 6, 24, 1, 1, 23, 13) (13, 13, 7, 12, 31, 26, 7, 31, 7, 22, 13, 8) C 36,24,5 M 36,24,1 (35, 16, 10, 11, 0, 33, 20, 23, 0, 2, 2, 4) C 36,24,9 M 36,24,4 (4, 28, 15, 7, 33, 26, 13, 19, 1, 1, 2, 21) (33, 9, 20, 4, 6, 32, 17, 24, 15, 12, 22, 10) C 38,24,5 M 38,24,1 (18, 5, 17, 23, 6, 5, 29, 1, 0, 0, 23, 21) (3, 33, 36, 1, 15, 11, 8, 12, 19, 17, 10, 33) C 38,24,6 M 38,24,1 (1, 22, 16, 12, 19, 14, 31, 19, 0, 0, 36, 24) (31, 5, 17, 28, 10, 21, 18, 11, 35, 25, 5, 20) C 38,24,7 M 38,24,1 (34, 16, 34, 7, 37, 2, 17, 19, 0, 0, 6, 32) (21, 4, 28, 35, 26, 34, 9, 30, 36, 11, 0, 26) C 38,24,8 M 38,24,3 (2, 22, 1, 1, 37, 5, 18, 9, 0, 1, 19, 35) (30, 11, 12, 23, 11, 32, 24, 17, 29, 35, 19, 33) C 38,24,9 M 38,24,3 (4, 28, 24, 35, 37, 13, 3, 34, 0, 0, 28, 12) (30, 18, 10, 10, 13, 32, 15, 36, 7, 6, 23, 32) C 40,24,5 M 40,24,1 (11, 26, 7, 16, 35, 33, 17, 9, 0, 1, 7, 2) (24, 36, 27, 6, 1, 1, 34, 18, 10, 7, 16, 34) C 40,24,6 M 40,24,1 (12, 2, 17, 30, 8, 28, 22, 39, 0, 0, 9, 17) (32, 38, 35, 4, 28, 9, 30, 39, 19, 30, 12, 20) C 40,24,7 M 40,24,2 (19, 20, 22, 15, 5, 23, 15, 9, 0, 1, 10, 33) (3, 1, 29, 34, 19, 21, 1, 1, 28, 15, 28, 6) C 40,24,8 M 40,24,3 (6, 39, 12, 39, 21, 27, 1, 32, 1, 1, 15, 34) (35, 29, 21, 13, 5, 37, 28, 14, 23, 1, 28, 34) C 40,24,9 M 40,24,3(23,16, 0, 28, 5, 9, 22, 22, 0, 2, 32, 27)(7, 29,11, 38, 30,21,8,12,26,10,22,24)…”

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confidence: 99%

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Construction of extremal Type II $\mathbb{Z}_{2k}$-codes

Harada¹

2022

Preprint

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We give methods for constructing many self-dual Z m -codes and Type II Z 2k -codes of length 2n starting from a given self-dual Z m -code and Type II Z 2k -code of length 2n, respectively. As an application, we construct extremal Type II Z 2k -codes of length 24 for k = 4, 5, . . . , 20 and extremal Type II Z 2k -codes of length 32 for k = 4, 5, . . . , 10. We also construct new extremal Type II Z 4 -codes of lengths 56 and 64.

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Section: Introductionmentioning

confidence: 92%

“…Remark 3.5. For the case k = 1, the above theorem can be found in [23]. Theorem 5 in [14] can be obtained from the above theorem by putting k = 2, x = t and y = (2, 2, .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Construction of extremal Type II $\mathbb{Z}_{2k}$-codes

Harada¹

2022

Preprint

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School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|•| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|•| 2 ) < ∞ for some ε > 0. RésuméDes conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de super-Poincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|•| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et étend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilité à support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|•| 2 ) < ∞ pour quelque ε > 0.

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Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme

Dougherty

Korban

Şahinkaya

et al. 2024

AMC

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<p style='text-indent:20px;'>We present a generator matrix of the form <inline-formula><tex-math id="M1">\begin{document}$ [ \sigma(v_1) \ | \ \sigma(v_2)] $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ v_1 \in RG $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ v_2\in RH $\end{document}</tex-math></inline-formula>, for finite groups <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> of order <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula> for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve <inline-formula><tex-math id="M8">\begin{document}$ 10 $\end{document}</tex-math></inline-formula> of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain <inline-formula><tex-math id="M9">\begin{document}$ 82 $\end{document}</tex-math></inline-formula> new binary linear complementary dual codes, <inline-formula><tex-math id="M10">\begin{document}$ 50 $\end{document}</tex-math></inline-formula> of which are either optimal or near optimal of lengths <inline-formula><tex-math id="M11">\begin{document}$ 41 \leq n \leq 61 $\end{document}</tex-math></inline-formula> which are new to the literature.</p>

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Construction for both self-dual codes and LCD codes

Cited by 3 publications

References 26 publications

Construction of extremal Type II $\mathbb{Z}_{2k}$-codes

Construction of extremal Type II $\mathbb{Z}_{2k}$-codes

School Mathematics Textbooks in China

Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme

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