Matrix-Product Codes over ? q

Blackmore, Tim; Norton, Graham H.

doi:10.1007/pl00004226

Cited by 91 publications

(95 citation statements)

References 6 publications

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“…We remark that this property also holds for the other constructions we study in this paper, but only when the C i 's are nested. Finally, we also study the squares of a matrix-product construction from [2] where we can apply the same proof techniques as we have in the other constructions.…”

Section: Results and Outlinementioning

confidence: 99%

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Squares of matrix-product codes

Cascudo

Gundersen²,

Ruano

2020

Finite Fields and Their Applications

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The component-wise or Schur product C * C ′ of two linear error-correcting codes C and C ′ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C ′ . When C = C ′ , we call the product the square of C and denote it C * 2 . Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrixproduct codes, a general construction that allows to obtain new longer codes from several "constituent" codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u, u+v)-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes).

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Section: Results and Outlinementioning

confidence: 99%

“…We recall the definition and basic properties of matrix-product codes (following [2]) and squares of codes. We call A = [a ij ] i=1,...,s,j=1,...,l the defining matrix and the C i 's the constituent codes.…”

Section: Preliminariesmentioning

confidence: 99%

Squares of matrix-product codes

Cascudo

Gundersen²,

Ruano

2020

Finite Fields and Their Applications

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“…If a 1 = 0, then a 2 ̸ = 0, and hence µ α (k) = µ α (b n a 2 ) ≥ n + µ α (a 2 ) ≥ n + ρ α (D (2) n ) ≥ n + ⌊β 2 n⌋ − t 2 + 1. 2.…”

Section: A Double M-constructionmentioning

confidence: 95%

“…Theorem 3.4 (Propagation Rule 10). Let P 1 be a (t 1 , α 1 , β 1 , n, m, s)-net in base b with dual set D (1) n and P 2 be a (t 2 , α 2 , β 2 , n, m, s)-net in base b with dual set D (2) n . Let d = d(D (1) n , D (2) n ).…”

Section: A Double M-constructionmentioning

confidence: 99%

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Duality theory and propagation rules for higher order nets

Baldeaux

Dick

Pillichshammer

2011

Discrete Mathematics

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a b s t r a c tHigher order nets and sequences are used in quasi-Monte Carlo rules for the approximation of high dimensional integrals over the unit cube. Hence one wants to have higher order nets and sequences of high quality.In this paper we introduce a duality theory for higher order nets whose construction is not necessarily based on linear algebra over finite fields. We use this duality theory to prove propagation rules for such nets. This way we can obtain new higher order nets (sometimes with improved quality) from existing ones. We also extend our approach to the construction of higher order sequences.

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