$ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-线性Hadamard码的递归构造

IF 0.7 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Advances in Mathematics of Communications Pub Date : 2023-01-01 DOI:10.3934/amc.2023047
Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva
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引用次数: 0

摘要

The $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-加性代码是的子组 $ \mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3} $. a $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $线性哈达玛码是一种哈达玛码,它是灰度图图像的一种 $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-附加代码。在本文中,我们推广了关于 $ \mathbb{Z}_2 \mathbb{Z}_4 $-线性Hadamard代码 $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-线性Hadamard代码与 $ \alpha_1 \neq 0 $, $ \alpha_2 \neq 0 $,和 $ \alpha_3 \neq 0 $. 首先,我们给出的递归构造 $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-加性Hadamard类型代码 $ (\alpha_1, \alpha_2, \alpha_3;t_1, t_2, t_3) $ 有 $ t_1\geq 1 $, $ t_2 \geq 0 $,和 $ t_3\geq 1 $. 众所周知,每一个 $ \mathbb{Z}_4 $线性Hadamard代码相当于a $ \mathbb{Z}_2 \mathbb{Z}_4 $线性Hadamard代码 $ \alpha_1\neq 0 $ 和 $ \alpha_2\neq 0 $. 不像 $ \mathbb{Z}_2 \mathbb{Z}_4 $-线性哈达玛码,一般来说,这个族的 $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-线性Hadamard代码不包括族 $ \mathbb{Z}_4 $-线性或 $ \mathbb{Z}_8 $-线性Hadamard代码。例如,我们展示了长度 $ 2^{11} $,构造的非线性 $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $线性哈达玛码彼此不等价,也不等价于任何 $ \mathbb{Z}_2 \mathbb{Z}_4 $-线性Hadamard,也不是任何先前构建的 $ \mathbb{Z}_{2^s} $-Hadamard code, with $ s\geq 2 $. 最后,我们也给出了的其他递归构造 $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-具有相同类型的加性Hadamard码,我们证明,应用Gray映射后,得到的码与之前的码是等价的。
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On recursive constructions of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes
The $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive codes are subgroups of $ \mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3} $. A $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard code is a Hadamard code, which is the Gray map image of a $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive code. In this paper, we generalize some known results for $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard codes to $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes with $ \alpha_1 \neq 0 $, $ \alpha_2 \neq 0 $, and $ \alpha_3 \neq 0 $. First, we give a recursive construction of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive Hadamard codes of type $ (\alpha_1, \alpha_2, \alpha_3;t_1, t_2, t_3) $ with $ t_1\geq 1 $, $ t_2 \geq 0 $, and $ t_3\geq 1 $. It is known that each $ \mathbb{Z}_4 $-linear Hadamard code is equivalent to a $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard code with $ \alpha_1\neq 0 $ and $ \alpha_2\neq 0 $. Unlike $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard codes, in general, this family of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes does not include the family of $ \mathbb{Z}_4 $-linear or $ \mathbb{Z}_8 $-linear Hadamard codes. We show that, for example, for length $ 2^{11} $, the constructed nonlinear $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes are not equivalent to each other, nor to any $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard, nor to any previously constructed $ \mathbb{Z}_{2^s} $-Hadamard code, with $ s\geq 2 $. Finally, we also present other recursive constructions of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.
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来源期刊
Advances in Mathematics of Communications
Advances in Mathematics of Communications 工程技术-计算机:理论方法
CiteScore
2.20
自引率
22.20%
发文量
78
审稿时长
>12 weeks
期刊介绍: Advances in Mathematics of Communications (AMC) publishes original research papers of the highest quality in all areas of mathematics and computer science which are relevant to applications in communications technology. For this reason, submissions from many areas of mathematics are invited, provided these show a high level of originality, new techniques, an innovative approach, novel methodologies, or otherwise a high level of depth and sophistication. Any work that does not conform to these standards will be rejected. Areas covered include coding theory, cryptology, combinatorics, finite geometry, algebra and number theory, but are not restricted to these. This journal also aims to cover the algorithmic and computational aspects of these disciplines. Hence, all mathematics and computer science contributions of appropriate depth and relevance to the above mentioned applications in communications technology are welcome. More detailed indication of the journal''s scope is given by the subject interests of the members of the board of editors.
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