On full-rank perfect codes over finite fields
نویسندگان
چکیده
منابع مشابه
Full-Rank Perfect Codes over Finite Fields
In this paper, we propose a construction of fullrank q-ary 1-perfect codes over finite fields. This construction is a generalization of the Etzion and Vardy construction of fullrank binary 1-perfect codes (1994). Properties of i-components of q-ary Hamming codes are investigated and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1...
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The paper deals with the perfect 1-error correcting codes over a finite field with q elements (briefly q-ary 1-perfect codes). We show that the orthogonal code to the q-ary non-full-rank 1-perfect code of length n = (q − 1)/(q − 1) is a q-ary constant-weight code with Hamming weight equals to qm−1 where m is any natural number not less than two. We derive necessary and sufficient conditions for...
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Golomb and Welch conjectured in 1970 that there only exist perfect Lee codes for radius t = 1 or dimension n = 1, 2. It is admitted that the existence and the construction of quasi-perfect Lee codes have to be studied since they are the best alternative to the perfect codes. In this paper we firstly highlight the relationships between subset sums, Cayley graphs, and Lee linear codes and present...
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On the structure of non-full-rank perfect codes Denis S. Krotov Sobolev Institute of Mathematics and Mechanics and Mathematics Department, Novosibirsk State University Novosibirsk, Russia The Krotov combining construction of perfect 1-error-correcting binary codes from 2000 and a theorem of Heden saying that every non-full-rank perfect 1-errorcorrecting binary code can be constructed by this co...
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Let F be a field. Given a simple graph G on n vertices, its minimal rank (with respect to F ) is the minimum rank of a symmetric n× n F -valued matrix whose off-diagonal zeroes are the same as in the adjacency matrix of G. If F is finite, then for every k, it is shown that the set of graphs of minimal rank at most k is characterized by finitely many forbidden induced subgraphs, each on at most ...
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ژورنال
عنوان ژورنال: Journal of Applied and Industrial Mathematics
سال: 2016
ISSN: 1990-4789,1990-4797
DOI: 10.1134/s1990478916030157