Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in C n × Q 8

Acevedo, Santiago Barrera; Dietrich, Heiko

doi:10.1007/s12095-017-0224-y

“…Also, they completed the computational results obtained by Baliga and Horadam. Barrera and Dietrich [6] proved that there is a 1-1 correspondence between the perfect sequences of length t over Q ∩ qQ, with q = (1 + i + j + k)/2, and the (4t, 2, 4t, 2t)-relative difference sets in C t × Q relative to C 2 .…”

Section: Hfp(2t 4 U )-Codesmentioning

confidence: 99%

On Hadamard full propelinear codes with associated group C_{2t}\times C_2

Bailera¹,

Borges

²

,

Rifà³

2021

Advances in Mathematics of Communications

0

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We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is C 2t × C 2. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by 3 if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order 16.

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“…Also, they completed the computational results obtained by Baliga and Horadam. Barrera and Dietrich [6] proved that there is a 1-1 correspondence between the perfect sequences of length t over Q ∩ qQ, with q = (1 + i + j + k)/2, and the (4t, 2, 4t, 2t)-relative difference sets in C t × Q relative to C 2 . iv) a = (A 1 , A 2 , .…”

Section: Hfp(t Q U )-Codesmentioning

confidence: 99%

Hadamard full propelinear codes with associated group $C_{2t}\times C_2$; rank and kernel

Bailera,

Borges,

Rifà

2018

Preprint

0

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We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is C 2t × C 2 . We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by 3 if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order 16.

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New families of quaternionic Hadamard matrices

Barrera Acevedo,

Dietrich,

Lionis

2024

Des. Codes Cryptogr.

0

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A quaternionic Hadamard matrix (QHM) of order n is an $$n\times n$$ n × n matrix H with non-zero entries in the quaternions such that $$HH^*=nI_n$$ H H ∗ = n I n , where $$I_n$$ I n and $$H^*$$ H ∗ denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.

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Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in C n × Q 8

Cited by 8 publications

References 18 publications

On Hadamard full propelinear codes with associated group C_{2t}\times C_2

On Hadamard full propelinear codes with associated group C_{2t}\times C_2

Hadamard full propelinear codes with associated group $C_{2t}\times C_2$; rank and kernel

New families of quaternionic Hadamard matrices

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