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Roger Tenvalls - One of the best experts on this subject based on the ideXlab platform.
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mathbb z _ 2 mathbb z _ 4 additive cyclic Codes generator polynomials and Dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb {Z}_{4}[x]$ -module $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . The parameters of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are determined in terms of the generator polynomials of the Code ${\mathcal{ C}}$ .
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z2z4 additive cyclic Codes generator polynomials and Dual Codes
arXiv: Discrete Mathematics, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive Code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic Code are determined in terms of the generator polynomials of the Code ${\cal C}$.
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z2z4 additive cyclic Codes generator polynomials and Dual Codes
Proceedings of Karatekin Mathematics Days, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A Z₂Z₄-additive Code C ⊆ Zα2 × Zβ₄ is called cyclic Code if the set of coordinates can be partitioned into two subsets, the set of Z₂ and the set of Z₄ coordinates, such that any cyclic shift of the coordinates of both subsets leaves invariant the Code. These Codes can be identified as submodules of the Z₄[x]-module Z₂[x]/(x^α − 1) × Z₄ [x]/(x^β − 1). The parameters of a Z₂Z₄-additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a Z₂Z₄-additive cyclic Code are determined in terms of the generator polynomials of the Code C.
Joaquim Borges - One of the best experts on this subject based on the ideXlab platform.
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mathbb z _ 2 mathbb z _ 4 additive cyclic Codes generator polynomials and Dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb {Z}_{4}[x]$ -module $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . The parameters of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are determined in terms of the generator polynomials of the Code ${\mathcal{ C}}$ .
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${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -Additive Cyclic Codes, Generator Polynomials, and Dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernández-córdoba, Roger Ten-vallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb {Z}_{4}[x]$ -module $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . The parameters of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are determined in terms of the generator polynomials of the Code ${\mathcal{ C}}$ .
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z2z4 additive cyclic Codes generator polynomials and Dual Codes
arXiv: Discrete Mathematics, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive Code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic Code are determined in terms of the generator polynomials of the Code ${\cal C}$.
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z2z4 additive cyclic Codes generator polynomials and Dual Codes
Proceedings of Karatekin Mathematics Days, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A Z₂Z₄-additive Code C ⊆ Zα2 × Zβ₄ is called cyclic Code if the set of coordinates can be partitioned into two subsets, the set of Z₂ and the set of Z₄ coordinates, such that any cyclic shift of the coordinates of both subsets leaves invariant the Code. These Codes can be identified as submodules of the Z₄[x]-module Z₂[x]/(x^α − 1) × Z₄ [x]/(x^β − 1). The parameters of a Z₂Z₄-additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a Z₂Z₄-additive cyclic Code are determined in terms of the generator polynomials of the Code C.
Cristina Fernandezcordoba - One of the best experts on this subject based on the ideXlab platform.
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mathbb z _ 2 mathbb z _ 4 additive cyclic Codes generator polynomials and Dual Codes
IEEE Transactions on Information Theory, 2016Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive Code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb {Z}_{4}[x]$ -module $\mathbb {Z}_{2}[x]/(x^\alpha -1)\times \mathbb {Z}_{4}[x]/(x^\beta -1)$ . The parameters of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive cyclic Code are determined in terms of the generator polynomials of the Code ${\mathcal{ C}}$ .
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z2z4 additive cyclic Codes generator polynomials and Dual Codes
arXiv: Discrete Mathematics, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive Code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of ${\mathbb{Z}}_4$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the Code invariant. These Codes can be identified as submodules of the $\mathbb{Z}_4[x]$-module $\mathbb{Z}_2[x]/(x^\alpha-1)\times\mathbb{Z}_4[x]/(x^\beta-1)$. The parameters of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic Code are determined in terms of the generator polynomials of the Code ${\cal C}$.
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z2z4 additive cyclic Codes generator polynomials and Dual Codes
Proceedings of Karatekin Mathematics Days, 2014Co-Authors: Joaquim Borges, Cristina Fernandezcordoba, Roger TenvallsAbstract:A Z₂Z₄-additive Code C ⊆ Zα2 × Zβ₄ is called cyclic Code if the set of coordinates can be partitioned into two subsets, the set of Z₂ and the set of Z₄ coordinates, such that any cyclic shift of the coordinates of both subsets leaves invariant the Code. These Codes can be identified as submodules of the Z₄[x]-module Z₂[x]/(x^α − 1) × Z₄ [x]/(x^β − 1). The parameters of a Z₂Z₄-additive cyclic Code are stated in terms of the degrees of the generator polynomials of the Code. The generator polynomials of the Dual Code of a Z₂Z₄-additive cyclic Code are determined in terms of the generator polynomials of the Code C.
Hsiechia Chang - One of the best experts on this subject based on the ideXlab platform.
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a 40 nm 535 mbps multiple Code rate turbo deCoder chip using reciprocal Dual trellis
IEEE Journal of Solid-state Circuits, 2013Co-Authors: Chenyang Lin, Chengchi Wong, Hsiechia ChangAbstract:This paper presents a multiple Code-rate turbo deCoder using the reciprocal Dual trellis to improve the hardware efficiency. For a convolutional Code with Code rate k/(k+1), its corresponding reciprocal Dual Code with rate 1/(k+1) has smaller Codeword space than the original Code while k > 1, leading to a simplified trellis of the high Code-rate Code. The proposed deCoder architecture can deCode Code rate k/(k+1) constituent convolutional Codes for k=1, 2, 4, 8, and 16. Moreover, two parallel soft-in/soft-out (SISO) deCoders are exploited in our turbo deCoder by using the quadratic permutation polynomial (QPP) interleaver to improve the decoding speed. After fabricated in 1P9M CMOS 40 nm process, the proposed deCoder with 1.27 mm2 core area can achieve 535 Mbps throughput at 8/9 Code rate, and the energy efficiency is 0.068 nJ/bit/iteration at 0.9 V.
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a 40 nm 535 mbps multiple Code rate turbo deCoder chip using reciprocal Dual trellis
Asian Solid-State Circuits Conference, 2012Co-Authors: Chenyang Lin, Chengchi Wong, Hsiechia ChangAbstract:This paper presents a turbo deCoder chip which can deCode Code rate k/(k + 1) constituent convolutional Codes for k =1, 2, 4, 8, and 16. After replacing the constituent Code by its Code rate 1/(k + 1) reciprocal Dual Code, we can derive a smaller Codeword space and design a simpler decoding trellis structure for high Code-rate SISO deCoder. In addition, two parallel SISO deCoders are exploited in our turbo deCoder by using the quadratic permutation polynomial (QPP) interleaver to improve the decoding speed. After fabricated in 1P9M CMOS 40 nm process, the proposed deCoder with 1.27 mm2 core area can achieve 535 Mbps throughput at 8/9 Code rate, and the energy efficiency is 0.068 nJ/bit/iteration at 0.9 V.
Vijay P Kumar - One of the best experts on this subject based on the ideXlab platform.
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a tight rate bound and a matching construction for locally recoverable Codes with sequential recovery from any number of multiple erasures
International Symposium on Information Theory, 2017Co-Authors: S B Balaji, Ganesh R Kini, Vijay P KumarAbstract:An [n, fc] Code C is said to be locally recoverable in the presence of a single erasure, and with locality parameter r, if each of the n Code symbols of C can be recovered by accessing at most r other Code symbols. An [n, k] Code is said to be a locally recoverable Code with sequential recovery from t erasures, if for any set of s ≤ t erasures, there is an s-step sequential recovery process, in which at each step, a single erased symbol is recovered by accessing at most r other Code symbols. This is equivalent to the requirement that for any set of s ≤ t erasures, the Dual Code contain a Codeword whose support contains the coordinate of precisely one of the s erased symbols. In this paper, a tight upper bound on the rate of such a Code, for any value of number of erasures t and any value r ≥ 3, of the locality parameter is derived. This bound proves an earlier conjecture due to Song, Cai and Yuen. While the bound is valid irrespective of the field over which the Code is defined, a matching construction of binary Codes that are rate-optimal is also provided, again for any value of t and any value r ≥ 3.
