Hadamard full propelinear codes of type Q; rank and kernel

J. Rifà; Emilio Suárez Canedo

doi:10.1007/s10623-017-0429-2

Hadamard full propelinear codes of type Q; rank and kernel

J. Rifà, Emilio Suárez Canedo

Research output: Contribution to journal › Article › Research › peer-review

1 Citation (Scopus)

Abstract

© 2017, Springer Science+Business Media, LLC. Hadamard full propelinear codes (HFP -codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of HFP -codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP -code of length 4n, dimension of the kernel k= 2 , and maximum rank r= 2 n, we obtain an HFP -code of double length 8n, dimension of the kernel k= 2 , and maximum rank r= 4 n.

Original language	English
Pages (from-to)	1905-1921
Journal	Designs, Codes, and Cryptography
Volume	86
Issue number	9
DOIs	https://doi.org/10.1007/s10623-017-0429-2
Publication status	Published - 1 Sep 2018

Keywords

Cocyclic Hadamard matrix
Hadamard code
Hadamard group
Kernel
Propelinear code
Rank
Relative difference set

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@article{460167d49bbd4fb69b0afa4cffb01cd4,

title = "Hadamard full propelinear codes of type Q; rank and kernel",

abstract = "{\textcopyright} 2017, Springer Science+Business Media, LLC. Hadamard full propelinear codes (HFP -codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of HFP -codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP -code of length 4n, dimension of the kernel k= 2 , and maximum rank r= 2 n, we obtain an HFP -code of double length 8n, dimension of the kernel k= 2 , and maximum rank r= 4 n.",

keywords = "Cocyclic Hadamard matrix, Hadamard code, Hadamard group, Kernel, Propelinear code, Rank, Relative difference set",

author = "J. Rif{\`a} and {Su{\'a}rez Canedo}, Emilio",

year = "2018",

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doi = "10.1007/s10623-017-0429-2",

language = "English",

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TY - JOUR

T1 - Hadamard full propelinear codes of type Q; rank and kernel

AU - Rifà, J.

AU - Suárez Canedo, Emilio

PY - 2018/9/1

Y1 - 2018/9/1

N2 - © 2017, Springer Science+Business Media, LLC. Hadamard full propelinear codes (HFP -codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of HFP -codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP -code of length 4n, dimension of the kernel k= 2 , and maximum rank r= 2 n, we obtain an HFP -code of double length 8n, dimension of the kernel k= 2 , and maximum rank r= 4 n.

AB - © 2017, Springer Science+Business Media, LLC. Hadamard full propelinear codes (HFP -codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of HFP -codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP -code of length 4n, dimension of the kernel k= 2 , and maximum rank r= 2 n, we obtain an HFP -code of double length 8n, dimension of the kernel k= 2 , and maximum rank r= 4 n.

KW - Cocyclic Hadamard matrix

KW - Hadamard code

KW - Hadamard group

KW - Kernel

KW - Propelinear code

KW - Rank

KW - Relative difference set

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DO - 10.1007/s10623-017-0429-2

M3 - Article

VL - 86

SP - 1905

EP - 1921

IS - 9

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