arXiv subjects

Mercè Villanueva

Publications and source records attributed to Mercè Villanueva.

17 recordsLinked to original sources

arXiv2025

Weight distributions of simplex codes over finite chain rings and their Gray map images

A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over $R$. These codes can be seen as a generalization of the linear codes over $\Z_{p^s}$ with $p$ prime and $s \geq 1$. In this paper, we present the construction of linear simplex codes over $R$ and their corresponding $R$-linear simplex codes of type $α$ and $β$. Moreover, we show the fundamental parameters of these codes, including their minimum Hamming distance, as well as their complete weight distributions. We also study whether these simplex codes are optimal with respect to the Griesmer-type bound.

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arXiv2024

Linearity and Classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-Linear Hadamard Codes

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1} \times \mathbb{Z}_4^{α_2} \times \mathbb{Z}_8^{α_3}$. 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. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(α_1,α_2, α_3;t_1,t_2, t_3)$ with $α_1 \neq 0$, $α_2 \neq 0$, $α_3 \neq 0$, $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$ is known. 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 $α_1 \neq 0$, $α_2 \neq 0$, and $α_3 \neq 0$. First, we show for which types the corresponding $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^t$ are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for $3 \leq t \leq 11$, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, which are not equivalent to any other constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code, nor to any $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, nor to any previously constructed $\mathbb{Z}_{2^s}$-linear Hadamard code with $s\geq 2$, with the same length $2^t$.

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arXiv2024

Computing efficiently a parity-check matrix for Zps-additive codes

The Zps-additive codes of length n are subgroups of Zps^n , and can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s . In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general. A time complexity analysis is also shown.

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arXiv2023

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

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1} \times \mathbb{Z}_4^{α_2} \times \mathbb{Z}_8^{α_3}$, and can be seen as linear codes over $\mathbb{Z}_2$ when $α_2=α_3=0$, $\mathbb{Z}_4$-additive or $\mathbb{Z}_8$-additive codes when $α_1=α_3=0$ or $α_1=α_2=0$, respectively, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $α_3=0$. 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 $α_1 \neq 0$, $α_2 \neq 0$, and $α_3 \neq 0$. First, we give a recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(α_1,α_2, α_3;t_1,t_2, t_3)$ 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 $α_1 \neq 0$, $α_2 \neq 0$, and $α_3 \neq 0$. 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 $2^{11}$.

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arXiv2022

$\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-Additive Generalized Hadamard Codes

The $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_p^{α_1} \times \mathbb{Z}_{p^2}^{α_2} \times \cdots \times \mathbb{Z}_{p^s}^{α_s}$, and can be seen as linear codes over $\mathbb{Z}_p$ when $α_i=0$ for all $i \in \{2,\dots, s\}$, a $\mathbb{Z}_{p^s}$-additive code when $α_i=0$ for all $i \in \{1,\dots, s-1\}$ , or a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code when $s=2$, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$ and $s=2$. A $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive code. In this paper, we generalize some known results for $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear GH codes with $p$ prime and $s\geq 2$. First, we give a recursive construction of $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots \mathbb{Z}_{p^s}$-additive GH codes of type $(α_1,\dots,α_s;t_1,\dots,t_s)$ with $t_1\geq 1, t_2,\dots,t_{s-1}\geq 0$, and $t_s\geq1$. Then, we show for which types the corresponding $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear GH codes are nonlinear over $\mathbb{Z}_p$. We also compute the kernel and its dimension whenever they are nonlinear.

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arXiv2022

Equivalences among Z_{p^s}-linear Generalized Hadamard Codes

The $\Z_{p^s}$-additive codes of length $n$ are subgroups of $\Z_{p^s}^n$, and can be seen as a generalization of linear codes over $\Z_2$, $\Z_4$, or $\Z_{2^s}$ in general. A $\Z_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\Z_p$ which is the image of a $\Z_{p^s}$-additive code by a generalized Gray map. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some $\Z_{p^s}$-linear GH codes of length $p^t$ 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 $t=10$, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel).

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arXiv2022

Construction and Linearity of Z_pZ_{p^2}-Linear Generalized Hadamard Codes

The $\Z_p\Z_{p^2}$-additive codes are subgroups of $\Z_p^{α_1} \times \Z_{p^2}^{α_2}$, and can be seen as linear codes over $\Z_p$ when $α_2=0$, $\Z_{p^2}$-additive codes when $α_1=0$, or $\Z_2\Z_4$-additive codes when $p=2$. A $\Z_p\Z_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\Z_p$ which is the Gray map image of a $\Z_p\Z_{p^2}$-additive code. In this paper, we generalize some known results for $\Z_p\Z_{p^2}$-linear GH codes with $p=2$ to any $p\geq 3$ prime when $α_1 \neq 0$. First, we give a recursive construction of $\Z_p\Z_{p^2}$-additive GH codes of type $(α_1,α_2;t_1,t_2)$ with $t_1,t_2\geq 1$. Then, we show for which types the corresponding $\Z_p\Z_{p^2}$-linear GH codes are non-linear over $\Z_p$. Finally, according to some computational results, we see that, unlike $\Z_4$-linear GH codes, when $p\geq 3$ prime, the $\Z_{p^2}$-linear GH codes are not included in the family of $\Z_p\Z_{p^2}$-linear GH codes with $α_1\not =0$.

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arXiv2020

Rank and Kernel of $\mathbb{F}_p$-Additive Generalised Hadamard Codes

A subset of a vector space $\mathbb{F}_q^n$ is $K$-additive if it is a linear space over the subfield $K\subseteq \mathbb{F}_q$. Let $q=p^e$, $p$ prime, and $e>1$. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are $\mathbb{F}_p$-additive are established. For specific ranks and dimensions of the kernel within these bounds, $\mathbb{F}_p$-additive GH codes are constructed. Moreover, for the case $e=2$, it is shown that the given bounds are tight and it is possible to construct an $\mathbb{F}_p$-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.

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arXiv2019

On the Kernel of $\Z_{2^s}$-Linear Simplex and MacDonald Codes

The $\Z_{2^s}$-additive codes are subgroups of $\Z^n_{2^s}$, and can be seen as a generalization of linear codes over $\Z_2$ and $\Z_4$. A $\Z_{2^s}$-linear code is a binary code which is the Gray map image of a $\Z_{2^s}$-additive code. We consider $\Z_{2^s}$-additive simplex codes of type $α$ and $β$, which are a generalization over $\Z_{2^s}$ of the binary simplex codes. These $\Z_{2^s}$-additive simplex codes are related to the $\Z_{2^s}$-additive Hadamard codes. In this paper, we use this relationship to establish the kernel of their binary images, under the Gray map, the $\Z_{2^s}$-linear simplex codes. Similar results can be obtained for the binary Gray map image of $\Z_{2^s}$-additive MacDonald codes.

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arXiv2019

Classification of the Z2Z4-linear Hadamard codes and their automorphism groups

A $Z_2Z_4$-linear Hadamard code of length $α+2β=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $α$ binary coordinates and $β$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $α=0$ and $α\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $α=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $α\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.

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arXiv2018

On the Kernel of $\mathbb{Z}_{2^s}$-Linear Hadamard Codes

The $\mathbb{Z}_{2^s}$-additive codes are subgroups of $\mathbb{Z}^n_{2^s}$, and can be seen as a generalization of linear codes over $\mathbb{Z}_2$ and $\mathbb{Z}_4$. A $\mathbb{Z}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a $\mathbb{Z}_{2^s}$-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the $\mathbb{Z}_4$-linear Hadamard codes. In this paper, the kernel of $\mathbb{Z}_{2^s}$-linear Hadamard codes and its dimension are established for $s > 2$. Moreover, we prove that this invariant only provides a complete classification for some values of $t$ and $s$. The exact amount of nonequivalent such codes are given up to $t=11$ for any $s\geq 2$, by using also the rank and, in some cases, further computations.

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arXiv2016

Partial permutation decoding for binary linear and Z4-linear Hadamard codes

Permutation decoding is a technique which involves finding a subset $S$, called PD-set, of the permutation automorphism group of a code $C$ in order to assist in decoding. An explicit construction of $\left \lfloor{\frac{2^m-m-1}{1+m}} \right \rfloor$-PD-sets of minimum size $\left \lfloor{\frac{2^m-m-1}{1+m}} \right \rfloor + 1$ for partial permutation decoding for binary linear Hadamard codes $H_m$ of length $2^m$, for all $m\geq 4$, is described. Moreover, a recursive construction to obtain $s$-PD-sets of size $l$ for $H_{m+1}$ of length $2^{m+1}$, from a given $s$-PD-set of the same size for $H_m$, is also established. These results are generalized to find $s$-PD-sets for (nonlinear) binary Hadamard codes of length $2^m$, called $\mathbb{Z}_4$-linear Hadamard codes, which are obtained as the Gray map image of quaternary linear codes of length $2^{m-1}$.

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arXiv2015

Ranks and Kernels of Codes from Generalized Hadamard Matrices

The ranks and kernels of generalized Hadamard matrices are studied. It is proven that any generalized Hadamard matrix $H(q,λ)$ over $F_q$, $q>3$, or $q=3$ and $\gcd(3,λ)\not =1$, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed.

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arXiv2013

Quasi-cyclic Flexible Regenerating Codes

In a distributed storage environment, where the data is placed in nodes connected through a network, it is likely that one of these nodes fails. It is known that the use of erasure coding improves the fault tolerance and minimizes the redundancy added in distributed storage environments. The use of regenerating codes not only make the most of the erasure coding improvements, but also minimizes the amount of data needed to regenerate a failed node. In this paper, a new family of regenerating codes based on quasi-cyclic codes is presented. Quasi-cyclic flexible minimum storage regenerating (QCFMSR) codes are constructed and their existence is proved. Quasi-cyclic flexible regenerating codes with minimum bandwidth constructed from a base QCFMSR code are also provided. These codes not only achieve optimal MBR parameters in terms of stored data and repair bandwidth, but also for an specific choice of the parameters involved, they can be decreased under the optimal MBR point. Quasi-cyclic flexible regenerating codes are very interesting because of their simplicity and low complexity. They allow exact repair-by-transfer in the minimum bandwidth case and an exact pseudo repair-by-transfer in the MSR case, where operations are needed only when a new node enters into the system replacing a lost one.

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arXiv2013

Permutation decoding of Z2Z4-linear codes

An alternative permutation decoding method is described which can be used for any binary systematic encoding scheme, regardless whether the code is linear or not. Thus, the method can be applied to some important codes such as Z2Z4-linear codes, which are binary and, in general, nonlinear codes in the usual sense. For this, it is proved that these codes allow a systematic encoding scheme. As a particular example, this permutation decoding method is applied to some Hadamard Z2Z4-linear codes.

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arXiv2013

A realistic distributed storage system: the rack model

In a realistic distributed storage environment, storage nodes are usually placed in racks, a metallic support designed to accommodate electronic equipment. It is known that the communication (bandwidth) cost between nodes which are in the same rack is much lower than between nodes which are in different racks. In this paper, a new model, where the storage nodes are placed in two racks, is proposed and analyzed. Moreover, the two-rack model is generalized to any number of racks. In this model, the storage nodes have different repair costs depending on the rack where they are placed. A threshold function, which minimizes the amount of stored data per node and the bandwidth needed to regenerate a failed node, is shown. This threshold function generalizes the ones given for previous distributed storage models. The tradeoff curve obtained from this threshold function is compared with the ones obtained from the previous models, and it is shown that this new model outperforms the previous ones in terms of repair cost.

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arXiv2013

A realistic distributed storage system that minimizes data storage and repair bandwidth

In a realistic distributed storage environment, storage nodes are usually placed in racks, a metallic support designed to accommodate electronic equipment. It is known that the communication (bandwidth) cost between nodes within a rack is much lower than the communication (bandwidth) cost between nodes within different racks. In this paper, a new model, where the storage nodes are placed in two racks, is proposed and analyzed. In this model, the storage nodes have different repair costs to repair a node depending on the rack where they are placed. A threshold function, which minimizes the amount of stored data per node and the bandwidth needed to regenerate a failed node, is shown. This threshold function generalizes the threshold function from previous distributed storage models. The tradeoff curve obtained from this threshold function is compared with the ones obtained from the previous models, and it is shown that this new model outperforms the previous ones in terms of repair cost.

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