On squares of cyclic codes

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The square C ∗2 of a linear error correcting code C is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in C. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both C ∗2 and C. In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes C with a relatively large dimension of C and minimum distance of the square C ∗2 are discussed. In some cases, the constructions lead to codes C such that both C and C ∗2 simultaneously have the largest possible minimum distances for their length and dimensions.

OriginalsprogEngelsk
Artikelnummer8451926
TidsskriftI E E E Transactions on Information Theory
Vol/bind65
Udgave nummer2
Sider (fra-til)1034-1047
Antal sider14
ISSN0018-9448
DOI
StatusUdgivet - feb. 2019

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title = "On squares of cyclic codes",
abstract = "The square C ∗2 of a linear error correcting code C is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in C. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both C ∗2 and C. In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes C with a relatively large dimension of C and minimum distance of the square C ∗2 are discussed. In some cases, the constructions lead to codes C such that both C and C ∗2 simultaneously have the largest possible minimum distances for their length and dimensions.",
keywords = "Binary codes, Complexity theory, Cryptography, Linear codes, Protocols, Reed-Solomon codes",
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On squares of cyclic codes. / Cascudo, Ignacio.

I: I E E E Transactions on Information Theory, Bind 65, Nr. 2, 8451926, 02.2019, s. 1034-1047.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - On squares of cyclic codes

AU - Cascudo, Ignacio

PY - 2019/2

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AB - The square C ∗2 of a linear error correcting code C is the linear code spanned by the component-wise products of every pair of (non-necessarily distinct) words in C. Squares of codes have gained attention for several applications mainly in the area of cryptography, and typically in those applications, one is concerned about some of the parameters (dimension and minimum distance) of both C ∗2 and C. In this paper, motivated mostly by the study of this problem in the case of linear codes defined over the binary field, squares of cyclic codes are considered. General results on the minimum distance of the squares of cyclic codes are obtained, and constructions of cyclic codes C with a relatively large dimension of C and minimum distance of the square C ∗2 are discussed. In some cases, the constructions lead to codes C such that both C and C ∗2 simultaneously have the largest possible minimum distances for their length and dimensions.

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KW - Complexity theory

KW - Cryptography

KW - Linear codes

KW - Protocols

KW - Reed-Solomon codes

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