On Z8-linear Hadamard codes : rank and classification

The Z2s -additive codes are subgroups of ℤZn2s, and can be seen as a generalization of linear codes over ℤ2 and ℤ4. A Zs-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ℤs -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the ℤ4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of ℤ2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s=3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established

On recursive constructions of Z2Z4Z8-linear Hadamard codes

The Z2Z4Z8-additive codes are subgroups of Z α1 2 × Z α2 4 × Z α3 8 . A Z2Z4Z8-linear Hadamard code is a Hadamard code, which is the Gray map image of a Z2Z4Z8-additive code. In this paper, we generalize some known results for Z2Z4-linear Hadamard codes to Z2Z4Z8-linear Hadamard codes with α1 ̸= 0, α2 ̸= 0, and α3 ̸= 0. First, we give a recursive construction of Z2Z4Z8- additive Hadamard codes of type (α1, α2, α3;t1, t2, t3) with t1 ≥ 1, t2 ≥ 0, and t3 ≥ 1. It is known that each Z4-linear Hadamard code is equivalent to a Z2Z4-linear Hadamard code with α1 ̸= 0 and α2 ̸= 0. Unlike Z2Z4-linear Hadamard codes, in general, this family of Z2Z4Z8-linear Hadamard codes does not include the family of Z4-linear or Z8-linear Hadamard codes. We show that, for example, for length 211, the constructed nonlinear Z2Z4Z8-linear Hadamard codes are not equivalent to each other, nor to any Z2Z4-linear Hadamard, nor to any previously constructed Z2s -Hadamard code, with s ≥ 2. Finally, we also present other recursive constructions of Z2Z4Z8-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones

On the linearity and classification of Z_p^s-linear generalized Hadamard codes

Acord transformatiu CRUE-CSICZp^s-additive codes of length n are subgroups of (Zp^s)^n , and can be seen as a generalization of linear codes over Z2, Z4 , or Z2^s in general. A Zp^s-linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zp^s-additive code by a generalized Gray map. In this paper, we generalize some known results for Zp^s-linear GH codes with 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 Zp^s -additive GH codes of type (n; t 1 , . . . , t s ), we show for which types the corresponding Zp^s-linear GH codes of length p^t are nonlinear over Zp .For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for p ≥ 3 are different from the case with 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 ≥ 2; by using also the rank as an invariant in some specific cases

Linearity and classification of ZpZp^2-linear generalized Hadamard codes

The ZpZp2 -additive codes are subgroups of Zα1 p × Zα2 p2 , and can be seen as linear codes over Zp when α2 = 0, Zp2 -additive codes when α1 = 0, or Z2Z4-additive codes when p = 2. A ZpZp2 -linear generalized Hadamard (GH) code is a GH code over Zp which is the Gray map image of a ZpZp2 -additive code. Recursive constructions of ZpZp2 -additive GH codes of type (α1, α2;t1,t2) with t1,t2 ≥ 1 are known. In this paper, we generalize some known results for ZpZp2 -linear GH codes with p = 2 to any p ≥ 3 prime when α1 = 0, and then we compare them with the ones obtained when α1 = 0. First, we show for which types the corresponding ZpZp2 -linear GH codes are nonlinear over Zp. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike Z4-linear Hadamard codes, the Zp2 -linear GH codes are not included in the family of ZpZp2 - linear GH codes with α1 = 0 when p ≥ 3 prime. Indeed, there are some families with infinite nonlinear ZpZp2 -linearGH codes, where the codes are not equivalent to any Zps - linear GH code with s ≥ 2

On the constructions of ZpZp2-linear generalized Hadamard codes

Altres ajuts: acord transformatiu CRUE-CSICThe ZZ-additive codes are subgroups of Z ×Z , and can be seen as linear codes over Z when α=0, Z-additive codes when α=0, or ZZ-additive codes when p=2. A ZZ-linear generalized Hadamard (GH) code is a GH code over Z which is the Gray map image of a ZZ-additive code. In this paper, we generalize some known results for ZZ-linear GH codes with p=2 to any p≥3 prime when α≠0. First, we give a recursive construction of ZZ-additive GH codes of type (α,α;t,t) with t,t≥1. We also present many different recursive constructions of ZZ-additive GH codes having the same type, and show that we obtain permutation equivalent codes after applying the Gray map. Finally, according to some computational results, we see that, unlike Z-linear GH codes, when p≥3 prime, the Z-linear GH codes are not included in the family of ZZ-linear GH codes with α≠0. Indeed, we observe that the constructed codes are not equivalent to the Z-linear GH codes for any s≥2

Tutories de comunicacio intercultural per a la mobilitat internacional (alemany)

El projecte d´innovació docent que es presenta es porta a terme a la Facultat d´Economia i Empresa de la UB, i té com a objectiu desenvolupar la competència intercultural dels estudiants de mobilitat internacional (outgoers espanyols/catalans - incomers de païssos de parla alemanya) abans i durant la seva estada. Es presentaran les dues primeres fases del projecte, que consisteix en la realització de dues sessions presencials de tutories entre iguals i sensibilització i el treball en tàndems

On the equivalence of Z ps -linear generalized Hadamard codes

Altres ajuts: acords transformatius de la UABLinear codes of length n over Zps , p prime, called Zps -additive codes, can be seen as subgroups of Zn ps . A Zps -linear generalized Hadamard (GH) code is a GH code over Zp which is the image of a Zps -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for p ≥ 3 prime, we establish that some Zps -linear GH codes of length pt having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to t = 10 if p = 3 or t = 8 if p = 5, this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel

Construction and classification of Z₂s-linear Hadamard codes

The Z₂s-additive and Z₂Z₄-additive codes are subgroups of Z₂s^n and Z₂^α × Z₄^β, respectively. Both families can be seen as generalizations of linear codes over Z₂ and Z₄. A Z₂s-linear (resp. Z₂Z₄-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Z₂s-additive (resp. Z₂Z₄-additive) code. It is known that there are exactly ⌊(t−1)/2⌋ and ⌊t/2⌋ nonequivalent Z₂Z₄-linear Hadamard codes of length 2ᵗ, with α=0 and α≠0, respectively, for all t≥3. In this paper, new Z₂s-linear Hadamard codes are constructed for s>2, which are not equivalent to any Z₂Z₄-linear Hadamard code. Moreover, for each s>2, it is claimed that the new constructed nonlinear Z₂-linear Hadamard codes of length 2ᵗ are pairwise nonequivalent