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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... more
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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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.... more
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 $\alpha$ and $\beta$, 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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Abstract The Z 2 s -additive codes are subgroups of Z 2 s n , and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z 2 s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z 2 s -additive... more
Abstract The Z 2 s -additive codes are subgroups of Z 2 s n , and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z 2 s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z 2 s -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 Z 4 -linear Hadamard codes. However, when s > 2 , the dimension of the kernel of Z 2 s -linear Hadamard codes of length 2 t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for s = 3 . Moreover, it is shown 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.
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Construction and classification of <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsev...more
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A code C is Z2Z4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary... more
A code C is Z2Z4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). In this paper, the rank and dimension of the kernel for Z2Z4-linear codes, which are the corresponding binary codes of Z2Z4-additive codes, are studied. The possible values of these two parameters for Z2Z4-linear codes, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z2Z4-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z2Z4-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z2Z4-additive code for each possible pair (r,k) is given.
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We establish upper and lower bounds on the rank and the dimension of the kernel of perfect binary codes. We also establish some results on the structure of perfect codes.
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ABSTRACT A quaternary linear Hadamard code ${\mathcal{C}}$ is a code over ${\mathbb{Z}_4}$ such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code ${\mathcal{C}}$ of... more
ABSTRACT A quaternary linear Hadamard code ${\mathcal{C}}$ is a code over ${\mathbb{Z}_4}$ such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code ${\mathcal{C}}$ of length n is defined as ${{\rm PAut}(\mathcal{C}) = \{\sigma \in S_{n} : \sigma(\mathcal{C}) = \mathcal{C}\}}$ . In this paper, the order of the permutation automorphism group of a family of quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by computing the orbits of the action of ${{\rm PAut}(\mathcal{C})}$ on ${\mathcal{C}}$ and by giving the generators of the group. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of these codes is also computed.
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Department of Information and Communications Engineering, Universitat Auto`noma de Barcelona{Bernat.Gaston | Jaume.Pujol | Merce.Villanueva}@uab.catAbstractIn a distributed storage environment, where the data is placed in nodes connected... more
Department of Information and Communications Engineering, Universitat Auto`noma de Barcelona{Bernat.Gaston | Jaume.Pujol | Merce.Villanueva}@uab.catAbstractIn a distributed storage environment, where the data is placed in nodes connected through a network, it is likelythat one of these nodes fails. It is known that the use of erasure coding improves the fault tolerance and minimizesthe redundancy added in distributed storage environments. The use of regenerating codes not only make the most ofthe 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 flexibleminimum storage regenerating (QCFMSR) codes are constructed and their existence is proved. Quasi-cyclic flexibleregenerating codes with minimum bandwidth constructed from a base QCFMSR code are also provided. These codesnot only achieve optimal MBR parameters in terms of stored dat...
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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... more
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.
A subset of a vector space <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}^{n}$ </tex-math></inline-formula> is additive if it is a linear space over the field <inline-formula>... more
A subset of a vector space <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}^{n}$ </tex-math></inline-formula> is additive if it is a linear space over the field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$q=p^{e}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> prime, and <inline-formula> <tex-math notation="LaTeX">$e>1$ </tex-math></inline-formula>. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case <inline-formula> <tex-math notation="LaTeX">$e=2$ </tex-math></inline-formula>, it is shown that the given bounds are tight and it is possible to construct an 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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Z<inf>2</inf><tex>$s$</tex>-additive codes are subgroups of Z<sup>n</sup><inf>2</inf>s, and can be seen as a generalization of linear codes over Z<inf>2</inf>and... more
Z<inf>2</inf><tex>$s$</tex>-additive codes are subgroups of Z<sup>n</sup><inf>2</inf>s, and can be seen as a generalization of linear codes over Z<inf>2</inf>and Z<inf>4</inf>. A Z<inf>2</inf><tex>$s$</tex> -linear code is a binary code (not necessarily linear) which is the Gray map image of a Z<inf>2</inf><tex>$s$</tex> -additive code. We consider Z<inf>2</inf>s- additive simplex codes of type a and β, which are a generalization over <tex>$Z$</tex><inf>2</inf><tex>$s$</tex> of the binary simplex codes. These codes are related to the Z<inf>2</inf><tex>$s$</tex> -additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding <tex>$Z$</tex><inf>2</inf>s-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, Z<inf>2</inf><tex>$s$</tex> -additive MacDonald codes are defined for <tex>$s$</tex> > 2, and equivalent results are obtained.
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The <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2^{s}}$ </tex-math></inline-formula>-additive codes are subgroups of <inline-formula> <tex-math... more
The <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2^{s}}$ </tex-math></inline-formula>-additive codes are subgroups of <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}^{n}_{2^{s}}$ </tex-math></inline-formula>, and can be seen as a generalization of linear codes over <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula>. A <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2^{s}}$ </tex-math></inline-formula>-linear Hadamard code is a binary Hadamard code which is the Gray map image of a <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2^{s}}$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula>-linear Hadamard codes. However, when <inline-formula> <tex-math notation="LaTeX">$s > 2$ </tex-math></inline-formula>, the dimension of the kernel of <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{2^{s}}$ </tex-math></inline-formula>-linear Hadamard codes of length <inline-formula> <tex-math notation="LaTeX">$2^{t}$ </tex-math></inline-formula> only provides a complete classification for some values of <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>. In this paper, the rank of these codes is computed for <inline-formula> <tex-math notation="LaTeX">$s=3$ </tex-math></inline-formula>. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once <inline-formula> <tex-math notation="LaTeX">$t\geq 3$ </tex-math></inline-formula> is fixed. In this case, the number of nonequivalent such codes is also established.
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A general criterion to obtain <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-PD-sets of minimum size <inline-formula> <tex-math notation="LaTeX">$s+1$... more
A general criterion to obtain <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-PD-sets of minimum size <inline-formula> <tex-math notation="LaTeX">$s+1$ </tex-math></inline-formula> for partial permutation decoding, which enable correction up to <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> errors, for systematic codes over a finite field <inline-formula> <tex-math notation="LaTeX">${ {F}}_{q}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${ {Z}}_{4}$ </tex-math></inline-formula>-linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over <inline-formula> <tex-math notation="LaTeX">${ {F}}_{q}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${ {Z}}_{4}$ </tex-math></inline-formula>-linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula>-PD-sets of size <inline-formula> <tex-math notation="LaTeX">$s+1$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">${ {Z}}_{4}$ </tex-math></inline-formula>-linear Hadamard codes of type <inline-formula> <tex-math notation="LaTeX">$2^\gamma 4^\delta $ </tex-math></inline-formula>, for all <inline-formula> <tex-math notation="LaTeX">$\delta \geq 4$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$1< s\leq 2^\delta -3$ </tex-math></inline-formula>; and for <inline-formula> <tex-math notation="LaTeX">${ {Z}}_{4}$ </tex-math></inline-formula>-linear simplex codes of type <inline-formula> <tex-math notation="LaTeX">$4^{m}$ </tex-math></inline-formula>, for all <inline-formula> <tex-math notation="LaTeX">$m\geq 2$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$1< s\leq 2^{m+1}-3$ </tex-math></inline-formula>, are also provided.
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The intersection problem for Z 2 Z 4 -additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z 2 Z 4 -additive codes C 1 and C 2 of the same length, is... more
The intersection problem for Z 2 Z 4 -additive (extended and nonextended) perfect codes, i.e., which are the possibilities for the number of codewords in the intersection of two Z 2 Z 4 -additive codes C 1 and C 2 of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructed. For all these Z 2 Z 4 -additive codes C 1 and C 2 , the abelian group structure of the intersection codes C 1 n C 2 is characterized. The parameters of this abelian group structure corresponding to the intersection codes are computed and lower and upper bounds for these parameters are established. Finally, for all possible parameters between these bounds, constructions of codes with these parameters for their intersections are given.
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A code [FORMULA] is [FORMULA]-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of [FORMULA] by deleting the coordinates outside X (respectively, Y) is a binary linear code... more
A code [FORMULA] is [FORMULA]-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of [FORMULA] by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of [FORMULA]-additive codes under an extended Gray map are called [FORMULA]-linear codes. In this paper, the invariants for [FORMULA]-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of [FORMULA]-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a [FORMULA]-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a [FORMULA]-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a [FORMULA]-linear code for each possible pair (r, k) is given.
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En marzo 2013 la UAB firmo convenio con Coursera para la imparticion de cursos MOOC. Hasta la fecha (17.5.2014) la UAB ha impartido tres cursos, esta impartiendo un cuarto curso, tiene dos mas ofertados y dos pactados y en espera. En esta... more
En marzo 2013 la UAB firmo convenio con Coursera para la imparticion de cursos MOOC. Hasta la fecha (17.5.2014) la UAB ha impartido tres cursos, esta impartiendo un cuarto curso, tiene dos mas ofertados y dos pactados y en espera. En esta contribucion presentamos la experiencia de la UAB: La relacion con Coursera, gestion de la plataforma, como canalizamos las solicitudes de nuestros profesores, que ayuda se les proporciona a estos, controles de calidad, etc. Se analizan tambien los resultados obtenidos en los cursos ya finalizados.
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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... more
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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In this paper we present new error-correcting block codes for two-dimensional signals constellations, such as QAM (quadrature amplitude modulation). The special interest of these new block codes is that they can correct one error in each... more
In this paper we present new error-correcting block codes for two-dimensional signals constellations, such as QAM (quadrature amplitude modulation). The special interest of these new block codes is that they can correct one error in each component of the codewords, with only one redundant symbol. So, it gives us a transmission rate R = N?1 N , where N is the length of the block code. We also proof that these new block codes can be constructed for any Euclidean complex quadratic eld. Some examples and simulation results when we use an additive Gaussian Channel are given.
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ABSTRACT The intersection structure for Z(2)Z(4)-additive Hadamard codes is investigated. For any two of these codes C(1) and C(2), the Abelian group structure of the intersection C(1) boolean AND C(2) is characterized. The parameters of... more
ABSTRACT The intersection structure for Z(2)Z(4)-additive Hadamard codes is investigated. For any two of these codes C(1) and C(2), the Abelian group structure of the intersection C(1) boolean AND C(2) is characterized. The parameters of this Abelian group structure corresponding to the intersection codes are computed, establishing lower and upper bounds for them. Constructions are given of codes whose intersection has any parameters between these bounds. Finally, the intersection problem, i.e., what the possibilities are for the number of codewords in the intersection of two Z(2)Z(4)-additive Hadamard codes C(1) and C(2) being of the same length, is also studied. Lower and upper bounds for the intersection number are established and, for any value between these bounds, codes with this intersection value are constructed.
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Permutation decoding is a technique which involves finding a subset S , called PD-set, of the permutation automorphism group PAut(C ) of a code C in order to assist in decoding. A method to obtain s -PD-sets of size s+1s+1 for partial... more
Permutation decoding is a technique which involves finding a subset S , called PD-set, of the permutation automorphism group PAut(C ) of a code C in order to assist in decoding. A method to obtain s -PD-sets of size s+1s+1 for partial permutation decoding for the binary linear Hadamard codes HmHm of length 2m2m, for all m≥4m≥4 and 1<s≤⌊2m−m−11+m⌋, is described. Moreover, a recursive construction to obtain s -PD-sets of size s+1s+1 for Hm+1Hm+1 of length 2m+12m+1, from a given s -PD-set of the same size for the Hadamard code of half length HmHm is also established.
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... more
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...
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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... more
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 ...