Butson full propelinear codes

Armario, J. A.; Bailera, Iván; Egan, Ronan

doi:10.48550/arxiv.2010.06206

Cited by 1 publication

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The generalization of the construction is quite straightforward, but the proof that the codes are GH is different from the binary case. This result has also been obtained independently in [1] by using another approach and considering Butson Hadamard matrices.…”

Section: Construction Of Z P S -Additive Gh Codessupporting

confidence: 69%

“…In [11], another technique is used to obtain that the Z 2 s -linear codes H t 1 ,...,t s are Hadamard. For p ≥ 3 prime, the result is already proved in [1]. However, we include another proof which is different, and it is not a generalization of the ones given in [19] and [11] neither.…”

Section: Examplementioning

confidence: 92%

“…This linear code over Z p s has length n = 1 and cardinality p s . Thus, the code H 1,0,...,0 = Φ(H 1,0,...,0 ) over Z p has length N = p s−1 and cardinality p s = N p. Actually, H 1,0,...,0 = Φ(Z p s ) is the linear GH code over Z p of length p s−1 used to define the Gray map Φ, so it is generated by (1).…”

Section: Examplementioning

confidence: 99%

“…Note that Z p s -linear Hadamard codes are in fact a particular case of these two-weight codes. More recently, Z p s -linear GH codes have been constructed in [1] as images under the Gray map of Butson Hadamard codes, defined from Butson Hadamard matrices.…”

Section: Introductionmentioning

confidence: 99%

“…. , u s−1 ] p is the p-ary expansion of u, that is, u = s−1 i=0 u i p i with u i ∈ Z p ; and Y is the same matrix as in (1), that is, a matrix of size (s − 1) × p s−1 whose columns are all the vectors in Z s−1 p . The Gray map defined by (3) is a particular case of the one defined by (2), when we consider P = {c 0 , c 1 , .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the linearity and classification of $${\mathbb {Z}}_{p^s}$$-linear generalized hadamard codes

Bhunia

Fernández-Córdoba

Villanueva

2022

Des. Codes Cryptogr.

View full text Add to dashboard Cite

Abstract$${\mathbb {Z}}_{p^s}$$ Z p s -additive codes of length n are subgroups of $${\mathbb {Z}}_{p^s}^n$$ Z p s n , and can be seen as a generalization of linear codes over $${\mathbb {Z}}_2$$ Z 2 , $${\mathbb {Z}}_4$$ Z 4 , or $${\mathbb {Z}}_{2^s}$$ Z 2 s in general. A $${\mathbb {Z}}_{p^s}$$ Z p s -linear generalized Hadamard (GH) code is a GH code over $${\mathbb {Z}}_p$$ Z p which is the image of a $${\mathbb {Z}}_{p^s}$$ Z p s -additive code by a generalized Gray map. In this paper, we generalize some known results for $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes with $$p=2$$ p = 2 to any odd prime p. First, we show some results related to the generalized Carlet’s Gray map. Then, by using an iterative construction of $${\mathbb {Z}}_{p^s}$$ Z p s -additive GH codes of type $$(n;t_1,\ldots , t_s)$$ ( n ; t 1 , … , t s ) , we show for which types the corresponding $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t are nonlinear over $${\mathbb {Z}}_p$$ Z p . For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for $$p\ge 3$$ p ≥ 3 are different from the case with $$p=2$$ p = 2 . Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any $$p\ge 2$$ p ≥ 2 ; by using also the rank as an invariant in some specific cases.

show abstract

Section: Construction Of Z P S -Additive Gh Codessupporting

confidence: 69%

Section: Examplementioning

confidence: 92%

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations