Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding

Etzion, Tuvi; Wachter-Zeh, Antonia

doi:10.1109/tit.2018.2797183

Cited by 32 publications

(28 citation statements)

References 33 publications

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“…A comparison between the required alphabet size for a scalar linear solution, a vector solution, and a scalar nonlinear solution, of the same multicast network is an important problem. It was proved in [18], [19] that there are multicast networks on which a vector network coding solution with vectors of length over F q outperforms any scalar linear network coding solution, i.e. the scalar solution requires an alphabet of size q s , where q s > q .…”

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confidence: 99%

Grassmannian Codes with New Distance Measures for Network Coding

Etzion

Zhang

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this work we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts it is proved that codes in the Hamming scheme form a subfamily of the Grassmannian codes.

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confidence: 99%

Grassmannian Codes with New Distance Measures for Network Coding

Etzion

Zhang

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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“…, S a of F l q . From the literature on subspace codes [13]- [15], we know that if l ≥ 2t then there exist at least q l−t pairwise trivially intersecting t-dimensional subspaces in F l q . Hence if q ≥ a 1 l−t and provided l ≥ 2t it is possible to find pairwise trivially intersecting t-dimensional subspaces S 1 , S 2 , .…”

Section: Code Construction 1 Our Aim Is To Construct a Matrixmentioning

confidence: 99%

“…whereŝ i,j is the (i, j) th entry ofŜ. Since any 2 or fewer columns ofŜ are linearly independent then using Theorem 3 in [13] we have that the columns of any 2 or fewer block matrices among S 1 , S 2 , . .…”

Section: Code Construction 1 Our Aim Is To Construct a Matrixmentioning

confidence: 99%

Codes for Updating Linear Functions over Small Fields

Ghosh

Natarajan

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We consider a point-to-point communication scenario where the receiver intends to maintain a specific linear function of a message vector over a finite field. When the value of the message vector changes, which is modelled as a sparse update, the transmitter broadcasts a coded version of the modified message while the receiver uses this codeword and the current value of the linear function to update its contents. It is assumed that the transmitter has access to only the modified message and is unaware of the exact difference vector between the original and modified messages. Under the assumption that the difference vector is sparse and that its Hamming weight is at the most a known constant, the objective is to design a linear code with as small a codelength as possible that allows successful update of the linear function at the receiver. This problem is motivated by applications to distributed data storage systems. Recently, Prakash and Médard derived a lower bound on the codelength, which is independent of the size of the underlying finite field, and provided constructions that achieve this bound if the size of the finite field is sufficiently large. However, this requirement on the field size can be prohibitive for even moderate values of the system parameters. In this paper, we provide a field-size aware analysis of the function update problem, including a tighter lower bound on the codelength, and design codes that trade-off the codelength for a smaller field size requirement. We also show that the problem of designing codes for updating linear functions is related to functional index coding or generalized index coding. We first characterize the family of function update problems where linear coding can provide reduction in codelength compared to a naive transmission scheme. We then provide field-size dependent bounds on the optimal codelength, and construct coding schemes based on error correcting codes and subspace codes when the receiver maintains linear functions of striped message vector. These codes provide a trade-off between the codelength and the size of the operating finite field, and whenever the achieved codelengths equal those reported by Prakash and Médard the requirements on the size of the finite field are matched as well. Finally, for any given function update problem, we construct an equivalent functional index coding or generalized index coding problem such that any linear coding scheme is valid for the function update problem if and only if it is valid for the constructed functional index coding problem.The authors are with the

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“…It was proved in [33] that the code formed from the dual subspaces (of dimension n − k) of a subspace packing is exactly what is required for a scalar solution for a family of networks called the generalized combination networks. This family of networks was used in [31,32] to show that vector network coding outperforms scalar linear network coding on multicast networks. The interested reader is invited to look in these papers for the required definitions and the proofs of the mentioned results.…”

Section: Introductionmentioning

confidence: 99%

Subspace packings: constructions and bounds

Etzion

Kurz

Otal

et al. 2020

Des. Codes Cryptogr.

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The Grassmannian G q (n, k) is the set of all k-dimensional subspaces of the vector space F n q . Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are qanalogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian G q (n, k) also form a family of q-analogs of block designs and they are called subspace designs.In this paper, we examine one of the last families of q-analogs of block designs which was not considered before. This family called subspace packings is the q-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing t-(n, k, λ) q is a set S of k-subspaces from G q (n, k) such that each t-subspace of G q (n, t) is contained in at most λ elements of S. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.

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Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding

Cited by 32 publications

References 33 publications

Grassmannian Codes with New Distance Measures for Network Coding

Grassmannian Codes with New Distance Measures for Network Coding

Codes for Updating Linear Functions over Small Fields

Subspace packings: constructions and bounds

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