Search results for " Hamming codes"

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Binary Hamming codes and Boolean designs

2021

AbstractIn this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we…

Applied Mathematics010102 general mathematicsGalois theoryZero (complex analysis)0102 computer and information sciencesAutomorphism01 natural sciencesComputer Science ApplicationsCombinatoricsBlock designs Hamming codes Permutation automorphisms Weight distribution Subset sum problemPermutation010201 computation theory & mathematicsWeight distributionSettore MAT/03 - Geometria0101 mathematicsHamming weightHamming codeVector spaceMathematics
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On the weight distribution of perfect binary codes

2021

In this paper, we give a new proof of the closed-form formula for the weight distribution of a perfect binary single-error-correcting code.

Discrete mathematicsAlgebra and Number TheoryPerfect codes Binary codes Hamming codes Weight distribution.Hamming boundApplied MathematicsBinary numberTheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGESSettore MAT/05 - Analisi MatematicaWeight distributionCode (cryptography)Binary codeSettore MAT/03 - GeometriaHamming codeAnalysisMathematics
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