Results 11 to 20 of about 9,296 (263)
The linear codes of t-designs held in the Reed-Muller and Simplex codes [PDF]
Cryptography and Communications, 2021 A fascinating topic of combinatorics is $t$-designs, which have a very long history. The incidence matrix of a $t$-design generates a linear code over GF$(q)$ for any prime power $q$, which is called the linear code of the $t$-design over GF$(q)$. On the other hand, some linear codes hold $t$-designs for some $t \geq 1$. The purpose of this paper is to
Cunsheng Ding, Chunming Tang 0001
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Optimal $$(2,\delta )$$ locally repairable codes via punctured simplex codes
Designs, Codes and Cryptography, 2023 Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal $(2, δ)$-LRCs over $\mathbb{F}_q$ with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a ...
Yue Gao 0001 +4 more
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The Complete Hierarchical Locality of the Punctured Simplex Code [PDF]
2019 IEEE International Symposium on Information Theory (ISIT), 2019 AbstractThis paper presents a new alphabet-dependent bound for codes with hierarchical locality. Then, the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code. This list is used to show that these codes are optimal codes with hierarchical locality.
Hollanti, Camilla, Grezet, Matthias
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On the cosets of the simplex code
Discrete Mathematics, 1985 Weight and distance properties of cosets of the simplex code are investigated. Let \(S_ m\) be the \((2^ m-1\), m, \(2^{m-1})\) simplex code and S its extension. It is shown that the coset of the all-1 vector of the extended code S of type \((2^ m\), m, \(2^{m-1})\) is the only coset of maximum weight \((2^{m-1})\).
Tor Helleseth, Harold F. Mattson
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Essential idempotents and simplex codes
Journal of Algebra Combinatorics Discrete Structures and Applications, 2017 We define essential idempotents in group algebras and use them to prove that every mininmal abelian non-cyclic code is a repetition code.Also we use them to prove that every minimal abelian code is equivalent to a minimal cyclic code of the same length.Finally, we show that a binary cyclic code is simplex if and only if is of length of the form n = 2 k
Gladys Chalom +2 more
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On maximal cliques in the graph of simplex codes
Journal of Geometry, 2023 The induced subgraph of the corresponding Grassmann graph formed by simplex codes is considered. We show that this graph, as the Grassmann graph, contains two types of maximal cliques. For any two cliques of the first type there is a monomial linear automorphism transferring one of them to the other.
Kwiatkowski, Mariusz, Pankov, Mark
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On some batch code properties of the simplex code
Designs, Codes and Cryptography, 2022 The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our approach is to relate these properties to certain (old and new) additive problems in finite abelian groups.
Henk D. L. Hollmann +3 more
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DNA Code Design Based on the Cosets of Codes over
Mathematics, 2023 DNA code design is a challenging problem, and it has received great attention in the literature due to its applications in DNA data storage, DNA origami, and DNA computing. The primary focus of this paper is in constructing new DNA codes using the cosets
Adel N. Alahmadi +2 more
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Nuclear Engineering and Technology, 2022 A low-energy dielectric loaded accelerator with a non-uniform, multi-segment structure is studied and optimized. So far, no analytical solution is provided for such structures.
M. Nikbakht +2 more
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Optimal (
IEEE Access, 2020 Locally repairable codes (LRCs) are implemented in distributed storage systems (DSSs) due to their low repair overhead. A linear code $\mathcal {C}$ is said to have $(r,\delta)$ -locality if for each coordinate $i$ , there exists a punctured subcode ...
Qiang Fu, Ruihu Li, Sen Yang
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