Hadamard full propelinear codes of type Q; rank and kernel

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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“…The following lemma will be useful in some proofs in the next sections. It is an stronger version of [22,Proposition 6]. Proof.…”

Section: Associated Groupmentioning

confidence: 98%

“…Proof. In [22,Proposition 6], it is proved that C| s consists of two copies of a Hadamard code of length 2t. As we project the code C over the support of s, we have that s| s = u 2t .…”

Section: Associated Groupmentioning

confidence: 99%

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

Bailera¹,

Borges

Rifà³

2021

Advances in Mathematics of Communications

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“…Furthermore, C is called full propelinear [15] if the permutation π x has not any fixed coordinate when x = 0, x = 1; and if 1 ∈ C then π 1 = Id n . Definition 2.4.…”

Section: Propelinear Codesmentioning

confidence: 99%

“…A generalized Hadamard code, which is also full propelinear, is called generalized Hadamard full propelinear code (briefly, GHFP-code). In the binary case, we have the Hadamard full propelinear codes, they were introduced in [15] and their equivalence with Hadamard groups was proven.…”

Section: Propelinear Codesmentioning

confidence: 99%

Generalized Hadamard full propelinear codes

Armario¹,

Bailera²,

Egan³

2019

Preprint

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Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full propelinear codes turns out. We prove that their existence is equivalent to the existence of central relative (v, w, v, v/w)-difference sets. Moreover, some structural properties of these codes are studied and examples are provided.

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“…The following lemma will be useful in some proofs through the next sections. It is an improved version of a lemma which appears in [22]. Proposition 3.6.…”

Section: Preliminariesmentioning

confidence: 99%

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

Bailera,

Borges,

Rifà

2018

Preprint

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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.

show abstract

Hadamard full propelinear codes of type Q; rank and kernel

Cited by 6 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

Generalized Hadamard full propelinear codes

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

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