Z2Z4Z8\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8-Additive Hadamard Codes

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$, and can be seen as linear codes over $\mathbb{Z}_2$ when $\alpha_2=\alpha_3=0$, $\mathbb{Z}_4$-additive or $\mathbb{Z}_8$-additive codes when $\alpha_1=\alpha_3=0$ or $\alpha_1=\alpha_2=0$, respectively, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $\alpha_3=0$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. In this paper, we generalize some known results for $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes to $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, and $\alpha_3 \neq 0$. First, we give a recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(\alpha_1,\alpha_2, \alpha_3;t_1,t_2, t_3)$ with $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$. Then, we show that in general the $\mathbb{Z}_4$-linear, $\mathbb{Z}_8$-linear and $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes are not included in the family of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, and $\alpha_3 \neq 0$. Actually, we point out that none of these nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^{11}$ is equivalent to a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code of any other type, a $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, or a $\mathbb{Z}_{2^s}$-linear Hadamard code, with $s\geq 2$, of the same length $2^{11}$

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

Equivalences among Z2s -linear Hadamard codes

The material in this paper was presented in part at the 16th International Workshop on Algebraic and Combinatorial Coding Theory in Svetlogorsk (Kaliningrad region), Russia, 2018.The Z2s -additive codes are subgroups of Z2s n, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s -linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s -linear Hadamard codes of length 2t is established by giving the exact number of such codes

CLADAG 2021 BOOK OF ABSTRACTS AND SHORT PAPERS

The book collects the short papers presented at the 13th Scientific Meeting of the Classification and Data Analysis Group (CLADAG) of the Italian Statistical Society (SIS). The meeting has been organized by the Department of Statistics, Computer Science and Applications of the University of Florence, under the auspices of the Italian Statistical Society and the International Federation of Classification Societies (IFCS). CLADAG is a member of the IFCS, a federation of national, regional, and linguistically-based classification societies. It is a non-profit, non-political scientific organization, whose aims are to further classification research

Additive Combinatorics: A Menu of Research Problems

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.Comment: This August 2017 version incorporates feedback and updates from several colleague