Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva
{"title":"$ \\mathbb{Z}_2 \\mathbb{Z}_4 \\mathbb{Z}_8 $-线性Hadamard码的递归构造","authors":"Dipak K. Bhunia, Cristina Fernández-Córdoba, Mercè Villanueva","doi":"10.3934/amc.2023047","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"21 1","pages":"0"},"PeriodicalIF":0.7000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On recursive constructions of $ \\\\mathbb{Z}_2 \\\\mathbb{Z}_4 \\\\mathbb{Z}_8 $-linear Hadamard codes\",\"authors\":\"Dipak K. 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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 $. 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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.
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.