Note on Niederreiter-Xing's Propagation Rule for Linear Codes

Özbudak, Ferruh

doi:10.1007/s002000100091

Cited by 37 publications

(43 citation statements)

References 2 publications

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“…, 6. From this and by C 6 ⊇ C 5 ⊇ C 4 ⊇ C 3 ⊇ C 2 ⊇ C 1 ⊇ C 0 , we deduce that d ∈ {1, 2, 3,4,5,6,7,9,10,12,14,15,18,20,21,25,28,30,35,42, 49, 0}. Now, let l = 3.…”

Section: An Examplementioning

confidence: 82%

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On matrix-product structure of repeated-root constacyclic codes over finite fields

Cao

Chen

Dinh

et al. 2020

Discrete Mathematics

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“…, 6. From this and by C 6 ⊇ C 5 ⊇ C 4 ⊇ C 3 ⊇ C 2 ⊇ C 1 ⊇ C 0 , we deduce that d ∈ {1, 2, 3,4,5,6,7,9,10,12,14,15,18,20,21,25,28,30,35,42, 49, 0}. Now, let l = 3.…”

Section: An Examplementioning

confidence: 82%

“…Theorem 2.1 (cf. [6], [3] Theorem 1) The matrix-product code [C 1 , ..., C α ]·A given by a sequence of [n, k i , d i ]-linear codes C i over F p m and a full-rank matrix A is a linear code whose length is nβ, it has dimension α i=1 k i and minimum distance larger than or equal to…”

Section: Preliminariesmentioning

confidence: 99%

On matrix-product structure of repeated-root constacyclic codes over finite fields

Cao

Chen

Dinh

et al. 2020

Discrete Mathematics

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“…We now turn our attention to the minimum distance of C. Denote by A i the matrix consisting of the first i rows of A and let C A i be the linear code spanned by the rows in A i . From [21], we have the following result on the minimum distance.…”

Section: Preliminariesmentioning

confidence: 99%

“…Matrix-product codes, formalized in [2], is a generalization of some previously known code constructions, such as for example the (u, u + v)-construction, also known as Plotkin sum. Matrix-product codes have been studied in several works, including [2,16,17,21].…”

Section: Introductionmentioning

confidence: 99%

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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“…We consider the matrix-product construction [U, V ] · D which was already introduced in [5] and rediscovered in [4,15]. In [4, Theorem3.7] a lower bound for the minimum distance of such code is given when the matrix D has a certain property, namely non-singular by columns.…”

Section: A New Mc-eliece Schemementioning

confidence: 99%

Using Reed-Solomon codes in the (U | U + V ) construction and an application to cryptography

Márquez-Corbella

Tillich

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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In this paper we present a modification of Reed-Solomon codes that beats the Guruwami-Sudan 1 − √ R decoding radius of Reed-Solomon codes at low rates R. The idea is to choose Reed-Solomon codes U and V with appropriate rates in a (U | U + V ) construction and to decode them with the Koetter-Vardy soft information decoder. We suggest to use a slightly more general version of these codes (but which has the same decoding performances as the (U | U + V )-construction) for code-based cryptography, namely to build a McEliece scheme. The point is here that these codes not only perform nearly as well (or even better in the low rate regime) as Reed-Solomon codes, their structure seems to avoid the Sidelnikov-Shestakov attack which broke a previous McEliece proposal based on generalized Reed-Solomon codes.2. Secondly, once we have recovered the right part v of the codeword, we can get a word (y 1 | y 2 − v) which should match two copies of a same word u of U . We can model this decoding problem by having some soft information.

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Note on Niederreiter-Xing's Propagation Rule for Linear Codes

Cited by 37 publications

References 2 publications

On matrix-product structure of repeated-root constacyclic codes over finite fields

On matrix-product structure of repeated-root constacyclic codes over finite fields

Squares of matrix-product codes

Using Reed-Solomon codes in the (U | U + V ) construction and an application to cryptography

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