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Antonia Wachterzeh - One of the best experts on this subject based on the ideXlab platform.

  • vector Network coding based on subspace codes outperforms scalar linear Network coding
    IEEE Transactions on Information Theory, 2018
    Co-Authors: Tuvi Etzion, Antonia Wachterzeh
    Abstract:

    This paper considers vector Network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same Multicast Network. The Multicast Networks considered in this paper have one source with $h$ messages, and the vector solution is over a field of size $q$ with vectors of length $t$ . For a given Network, let the smallest field size for which the Network has a scalar linear solution be $q_{s}$ , then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be $q_{s}-q^{t}$ . In this contribution, the achieved gap is $q^{(h-2)t^{2}/h + o(t)}$ for any $q \geq 2$ and any even $h \geq 4$ . If $h \geq 5$ is odd, then the achieved gap of the alphabet size is $q^{(h-3)t^{2}/(h-1) + o(t)}$ . Previously, only a gap of size one had been shown for Networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear Network coding solution for Multicast Networks. For three messages, we also show an advantage of vector Network coding, while for two messages the problem remains open. Several Networks are considered, all of them are generalizations and modifications of the well-known combination Networks. The vector Network codes that are used as solutions for those Networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

  • vector Network coding based on subspace codes outperforms scalar linear Network coding
    International Symposium on Information Theory, 2016
    Co-Authors: Tuvi Etzion, Antonia Wachterzeh
    Abstract:

    This paper considers vector Network coding based on rank-metric codes and subspace codes. Our main result is that vector Network coding can significantly reduce the required field size compared to scalar linear Network coding in the same Multicast Network. The achieved gap between the field size of scalar and vector Network coding is in q(h−2)t2/h+o(t) for any q ≥ 2 and any even h ≥ 4, where t denotes the dimension of the vector solution and h the number of messages. If h ≥ 5 is odd, then the achieved gap of the field size between the scalar Network coding solution and the vector Network coding solution is q(h−3)t2/(h−1)+o(t). Previously, only a gap of constant size had been shown. This implies also the same gap between the field size in linear and non-linear scalar Network coding for Multicast Networks. The results are obtained by considering several Multicast Networks which are variations of the well-known combination Network.

  • vector Network coding based on subspace codes outperforms scalar linear Network coding
    arXiv: Information Theory, 2015
    Co-Authors: Tuvi Etzion, Antonia Wachterzeh
    Abstract:

    This paper considers vector Network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same Multicast Network. The Multicast Networks considered in this paper have one source with $h$ messages, and the vector solution is over a field of size $q$ with vectors of length~$t$. For a given Network, let the smallest field size for which the Network has a scalar linear solution be $q_s$, then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be $q_s-q^t$. In this contribution, the achieved gap is $q^{(h-2)t^2/h + o(t)}$ for any $q \geq 2$ and any even $h \geq 4$. If $h \geq 5$ is odd, then the achieved gap of the alphabet size is $q^{(h-3)t^2/(h-1) + o(t)}$. Previously, only a gap of size size one had been shown for Networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear Network coding solution for Multicast Networks. For three messages, we also show an advantage of vector Network coding, while for two messages the problem remains open. Several Networks are considered, all of them are generalizations and modifications of the well-known combination Networks. The vector Network codes that are used as solutions for those Networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

Tuvi Etzion - One of the best experts on this subject based on the ideXlab platform.

  • vector Network coding based on subspace codes outperforms scalar linear Network coding
    IEEE Transactions on Information Theory, 2018
    Co-Authors: Tuvi Etzion, Antonia Wachterzeh
    Abstract:

    This paper considers vector Network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same Multicast Network. The Multicast Networks considered in this paper have one source with $h$ messages, and the vector solution is over a field of size $q$ with vectors of length $t$ . For a given Network, let the smallest field size for which the Network has a scalar linear solution be $q_{s}$ , then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be $q_{s}-q^{t}$ . In this contribution, the achieved gap is $q^{(h-2)t^{2}/h + o(t)}$ for any $q \geq 2$ and any even $h \geq 4$ . If $h \geq 5$ is odd, then the achieved gap of the alphabet size is $q^{(h-3)t^{2}/(h-1) + o(t)}$ . Previously, only a gap of size one had been shown for Networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear Network coding solution for Multicast Networks. For three messages, we also show an advantage of vector Network coding, while for two messages the problem remains open. Several Networks are considered, all of them are generalizations and modifications of the well-known combination Networks. The vector Network codes that are used as solutions for those Networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

  • vector Network coding based on subspace codes outperforms scalar linear Network coding
    International Symposium on Information Theory, 2016
    Co-Authors: Tuvi Etzion, Antonia Wachterzeh
    Abstract:

    This paper considers vector Network coding based on rank-metric codes and subspace codes. Our main result is that vector Network coding can significantly reduce the required field size compared to scalar linear Network coding in the same Multicast Network. The achieved gap between the field size of scalar and vector Network coding is in q(h−2)t2/h+o(t) for any q ≥ 2 and any even h ≥ 4, where t denotes the dimension of the vector solution and h the number of messages. If h ≥ 5 is odd, then the achieved gap of the field size between the scalar Network coding solution and the vector Network coding solution is q(h−3)t2/(h−1)+o(t). Previously, only a gap of constant size had been shown. This implies also the same gap between the field size in linear and non-linear scalar Network coding for Multicast Networks. The results are obtained by considering several Multicast Networks which are variations of the well-known combination Network.

  • vector Network coding based on subspace codes outperforms scalar linear Network coding
    arXiv: Information Theory, 2015
    Co-Authors: Tuvi Etzion, Antonia Wachterzeh
    Abstract:

    This paper considers vector Network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same Multicast Network. The Multicast Networks considered in this paper have one source with $h$ messages, and the vector solution is over a field of size $q$ with vectors of length~$t$. For a given Network, let the smallest field size for which the Network has a scalar linear solution be $q_s$, then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be $q_s-q^t$. In this contribution, the achieved gap is $q^{(h-2)t^2/h + o(t)}$ for any $q \geq 2$ and any even $h \geq 4$. If $h \geq 5$ is odd, then the achieved gap of the alphabet size is $q^{(h-3)t^2/(h-1) + o(t)}$. Previously, only a gap of size size one had been shown for Networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear Network coding solution for Multicast Networks. For three messages, we also show an advantage of vector Network coding, while for two messages the problem remains open. Several Networks are considered, all of them are generalizations and modifications of the well-known combination Networks. The vector Network codes that are used as solutions for those Networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

Yuanyuan Yang - One of the best experts on this subject based on the ideXlab platform.

  • a linear inter session Network coding scheme for Multicast
    Journal of Communications, 2009
    Co-Authors: Min Yang, Yuanyuan Yang
    Abstract:

    Network coding is a promising generalization of routing which allows a Network node to generate output messages by encoding its received messages to reduce the bandwidth consumption in the Network. An important application where Network coding offers unique advantages is the Multicast Network where a source node generates messages and multiple receivers collect the messages. Previous Network coding schemes primarily considered encoding the messages in a single Multicast session. In this paper, we consider the linear inter-session Network coding for Multicast. The basic idea is to divide the sessions into different groups and construct a linear Network coding scheme for each group. We use two metrics to guide the group division: overlap ratio and overlap width. These two metrics measure the benefit that a system can achieve by inter-session Network coding with different considerations. The overlap ratio mainly characterizes the Network bandwidth while the overlap width characterizes the system throughput. Our simulation results show that the proposed inter-session Network coding scheme can achieve about 30% higher throughput than intra-session Network coding.

  • a linear inter session Network coding scheme for Multicast
    Network Computing and Applications, 2008
    Co-Authors: Min Yang, Yuanyuan Yang
    Abstract:

    Network coding is a promising generalization of routing which allows a Network node to generate output messages by encoding its received messages to reduce the bandwidth consumption in the Network. An important application where Network coding offers unique advantages is the Multicast Network where a source node generates messages and multiple receivers collect the messages. Previous Network coding schemes primarily considered encoding the messages in a single Multicast session. In this paper, we consider the linear inter-session Network coding for Multicast. The basic idea is to divide the sessions into different groups and construct a linear Network coding scheme for each group. To maximize the performance, we introduce two metrics: overlap ratio and overlap width, to measure the benefit that a system can achieve by inter-session Network coding. The overlap ratio mainly characterizes the Network bandwidth while the overlap width characterizes the system throughput. Our simulation results show that the proposed inter-session Network coding scheme can achieve about 30% higher throughput than intra-session Network coding.

  • a more accurate analytical model on blocking probability of Multicast Networks
    IEEE Transactions on Communications, 2000
    Co-Authors: Yuanyuan Yang, J Wang
    Abstract:

    Multicast communication is one of the most important collective communication operations and is highly demanded in telecommunication environments and scalable parallel and distributed computing systems. In this paper, we consider the issue of supporting Multicast in the widely used a three-stage Clos Network or /spl upsi/(m,n,r) Network. We improve a previously proposed analytical model (Yang and Wang 1998) for the blocking probability of the /spl upsi/(m,n,r) Multicast Network by introducing more reasonable assumptions based on the properties of Multicast communication and the Clos Network. We also compare the improved analytical model with the simulation results under three typical routing control strategies. As can be seen, the improved model matches better with the simulation results and further confirms that a /spl upsi/(m,n,r) Network with a comparable cost to a permutation Network is almost nonblocking for Multicast connections.

  • on blocking probability of Multicast Networks
    IEEE Transactions on Communications, 1998
    Co-Authors: Yuanyuan Yang, J Wang
    Abstract:

    Multicast is a vital operation in both broad-band integrated services digital Networks (BISDN) and scalable parallel computers. We look into the issue of supporting Multicast in the widely used three-stage Clos Network or /spl upsi/(m, n, r) Network. Previous work has shown that a nonblocking /spl upsi/(m, n, r) Multicast Network requires a much higher Network cost than a /spl upsi/(m, n, r) permutation Network. However, little has been known on the blocking behavior of the /spl upsi/(m, n, r) Multicast Network with only a comparable Network cost to a permutation Network. We first develop an analytical model for the blocking probability of the /spl upsi/(m, n, r) Multicast Network and then study the blocking behavior of the Network under various routing control strategies through simulations. Our analytical and simulation results show that a /spl upsi/(m, n, r) Network with a small number of middle switches m, such as m=n+c or dn, where c and d are small constants, is almost nonblocking for Multicast connections, although theoretically it requires m/spl ges//spl Theta/(n(log r/log log r)) to achieve nonblocking for Multicast connections. We also demonstrate that routing control strategies are effective for reducing the blocking probability of the Multicast Network. The best routing control strategy can provide a factor of two to three performance improvement over random routing. The results indicate that a /spl upsi/(m, n, r) Network with a comparable cost to a permutation Network can provide cost-effective support for Multicast communication.

Keping Long - One of the best experts on this subject based on the ideXlab platform.

  • circular shift linear Network codes with arbitrary odd block lengths
    IEEE Transactions on Communications, 2019
    Co-Authors: Qifu Tyler Sun, Hanqi Tang, Xiaolong Yang, Keping Long
    Abstract:

    Circular-shift linear Network coding (LNC) is a class of vector LNC with low encoding and decoding complexities, and with local encoding kernels chosen from cyclic permutation matrices. When $L$ is a prime with primitive root 2, it was recently shown that a scalar linear solution over GF( $2^{L-1}$ ) induces an $L$ -dimensional circular-shift linear solution at rate $(L-1)/L$ . In this paper, we prove that for arbitrary odd $L$ , every scalar linear solution over GF( $2^{m_{L}}$ ), where $m_{L}$ refers to the multiplicative order of 2 modulo $L$ , can induce an $L$ -dimensional circular-shift linear solution at a certain rate. Based on the generalized connection, we further prove that for such $L$ with $m_{L}$ beyond a threshold, every Multicast Network has an $L$ -dimensional circular-shift linear solution at rate $\phi (L)/L$ , where $\phi (L)$ is the Euler’s totient function of $L$ . An efficient algorithm for constructing such a solution is designed. Finally, we prove that every Multicast Network is asymptotically circular-shift linearly solvable.

  • circular shift linear Network coding
    IEEE Transactions on Information Theory, 2019
    Co-Authors: Hanqi Tang, Zongpeng Li, Xiaolong Yang, Keping Long
    Abstract:

    We study a class of linear Network coding (LNC) schemes, called circular-shift LNC, whose encoding operations consist of only circular-shifts and bit-wise additions. Formulated as a special vector linear code over GF(2), an $L$ -dimensional circular-shift linear code of degree $\delta $ restricts its local encoding kernels to be the summation of at most $\delta $ cyclic permutation matrices of size $L$ . We show that on a general Network, for a certain block length $L$ , every scalar linear solution over GF( $2^{L-1}$ ) can induce an $L$ -dimensional circular-shift linear solution with 1-bit redundancy per-edge transmission. Consequently, specific to a Multicast Network, such a circular-shift linear solution of an arbitrary degree $\delta $ can be efficiently constructed, which has an interesting complexity tradeoff between encoding and decoding with different choices of $\delta $ . By further proving that circular-shift LNC is insufficient to achieve the exact capacity of certain Multicast Networks, we show the optimality of the efficiently constructed circular-shift linear solution in the sense that its 1-bit redundancy is inevitable. Finally, both theoretical and numerical analysis imply that with increasing $L$ , a randomly constructed circular-shift linear code has linear solvability behavior comparable to a randomly constructed permutation-based linear code, but has shorter overheads.

Hanqi Tang - One of the best experts on this subject based on the ideXlab platform.

  • circular shift linear Network codes with arbitrary odd block lengths
    IEEE Transactions on Communications, 2019
    Co-Authors: Qifu Tyler Sun, Hanqi Tang, Xiaolong Yang, Keping Long
    Abstract:

    Circular-shift linear Network coding (LNC) is a class of vector LNC with low encoding and decoding complexities, and with local encoding kernels chosen from cyclic permutation matrices. When $L$ is a prime with primitive root 2, it was recently shown that a scalar linear solution over GF( $2^{L-1}$ ) induces an $L$ -dimensional circular-shift linear solution at rate $(L-1)/L$ . In this paper, we prove that for arbitrary odd $L$ , every scalar linear solution over GF( $2^{m_{L}}$ ), where $m_{L}$ refers to the multiplicative order of 2 modulo $L$ , can induce an $L$ -dimensional circular-shift linear solution at a certain rate. Based on the generalized connection, we further prove that for such $L$ with $m_{L}$ beyond a threshold, every Multicast Network has an $L$ -dimensional circular-shift linear solution at rate $\phi (L)/L$ , where $\phi (L)$ is the Euler’s totient function of $L$ . An efficient algorithm for constructing such a solution is designed. Finally, we prove that every Multicast Network is asymptotically circular-shift linearly solvable.

  • circular shift linear Network coding
    IEEE Transactions on Information Theory, 2019
    Co-Authors: Hanqi Tang, Zongpeng Li, Xiaolong Yang, Keping Long
    Abstract:

    We study a class of linear Network coding (LNC) schemes, called circular-shift LNC, whose encoding operations consist of only circular-shifts and bit-wise additions. Formulated as a special vector linear code over GF(2), an $L$ -dimensional circular-shift linear code of degree $\delta $ restricts its local encoding kernels to be the summation of at most $\delta $ cyclic permutation matrices of size $L$ . We show that on a general Network, for a certain block length $L$ , every scalar linear solution over GF( $2^{L-1}$ ) can induce an $L$ -dimensional circular-shift linear solution with 1-bit redundancy per-edge transmission. Consequently, specific to a Multicast Network, such a circular-shift linear solution of an arbitrary degree $\delta $ can be efficiently constructed, which has an interesting complexity tradeoff between encoding and decoding with different choices of $\delta $ . By further proving that circular-shift LNC is insufficient to achieve the exact capacity of certain Multicast Networks, we show the optimality of the efficiently constructed circular-shift linear solution in the sense that its 1-bit redundancy is inevitable. Finally, both theoretical and numerical analysis imply that with increasing $L$ , a randomly constructed circular-shift linear code has linear solvability behavior comparable to a randomly constructed permutation-based linear code, but has shorter overheads.