A class of non-linearly solvable networks

Connelly, Joseph; Zeger, K.

doi:10.1109/isit.2016.7541642

Cited by 6 publications

(30 citation statements)

References 22 publications

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“…Specifically, we ask if for any given edge e * the non-linear ( , R, n)-feasible network code presented in these works can be modified to give an ( , R, n)-feasible scheme in which the encoding on edge e * is CWL, implying that the CWL statement holds for the instance and rate vector under study. In the case studies from [14]- [16] we are able to modify the non-linear coding scheme appropriately, thus supporting the CWL statement. However we are not able to prove (or disprove) the same for the instance given in [17].…”

Section: On Proving the Edge-removal Statement Through The Local Lens...supporting

confidence: 65%

“…As any instance for which linear encoding functions are optimal satisfies the conditions of the question above, we study instances for which linear encoding functions are suboptimal. While we do not resolve the question in this work, we prove that such code modifications are possible for the network coding instances and solutions presented in [14]- [16] for which linear coding is known to be sub-optimal, implying that edge removal holds for these solutions. We note that our question is not resolved on the network instance given in [17] for which, as above, linear network coding is suboptimal.…”

Section: Introductionmentioning

confidence: 84%

“…C. The network instance of [16] In [16], a class of networks which is solvable but not linearly solvable is introduced. Using the notation of [16], a In B(m) the only encoding edges are the edges carrying message e 0 , e 1 , .…”

Section: Case Studies Of Statementmentioning

confidence: 99%

“…, e m , e are directly connected to sources such that the local and global encoding function on these edges are identical (as the incoming edge messages are the source messages). Now we show, for any edge in the network N 4 defined in [16] that we can modify the code of [16] and make the global encoding function φ ge CWL.…”

Section: Case Studies Of Statementmentioning

confidence: 99%

“…1) Subnetwork: N 1 (m): In the coding scheme of [16] all edges in N 1 (m) have a linear global encoding function.…”

Section: Case Studies Of Statementmentioning

confidence: 99%

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A Local Perspective on the Edge Removal Problem

Wei

Langberg

Effros

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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The edge removal problem studies the loss in network coding rates that results when a network communication edge is removed from a given network. It is known, for example, that in networks restricted to linear coding schemes and networks restricted to Abelian group codes, removing an edge e * with capacity Re * reduces the achievable rate on each source by no more than Re * . In this work, we seek to uncover larger families of encoding functions for which the edge removal statement holds. We take a local perspective: instead of requiring that all network encoding functions satisfy certain restrictions (e.g., linearity), we limit only the function carried on the removed edge e * . Our central results give sufficient conditions on the function carried by edge e * in the code used to achieve a particular rate vector under which we can demonstrate the achievability of a related rate vector once e * is removed.

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Section: On Proving the Edge-removal Statement Through The Local Lens...supporting

confidence: 65%

Section: Introductionmentioning

confidence: 84%

Section: Case Studies Of Statementmentioning

confidence: 99%

Section: Case Studies Of Statementmentioning

confidence: 99%

“…1) Subnetwork: N 1 (m): In the coding scheme of [16] all edges in N 1 (m) have a linear global encoding function.…”

Section: Case Studies Of Statementmentioning

confidence: 99%

See 3 more Smart Citations

A Local Perspective on the Edge Removal Problem

Wei

Langberg

Effros

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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A class of non-linearly solvable networks

Connelly

Zeger

2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths

Sun

Tang

et al. 2019

IEEE Trans. Commun.

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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 work, we prove that for arbitrary odd L, every scalar linear solution over GF(2 mL ), 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 φ(L)/L, where φ(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. DRAFTJanuary 3, 2019 1 Throughout this paper, the notion [Ae] e∈E ′ will always refer to column-wise juxtaposition of matrices Ae with e orderly chosen from a subset E ′ of E, where Ae may degenerate to vectors. DRAFT (26)For simplicity, writeTogether with (25), Eq. (26) can be written as U 1Ṽ + U 2 = I J . Because U T 1 andṼ can be respectively regarded as a Vandermonde matrix generated by α −j , j ∈ J and by α j , j ∈ J , they are invertible, and so is I J + U 2 . Thus, C. Proof of Theorem 4It remains to prove (11). Follow the same argument as in the proof of Theorem 1 (refer to Appendix-A) till Eq. (24). For a receiver t, by (21), (22) and (24), we have [F e ] e∈In(t) = (I ω ⊗ V L )P TM P(I ω ⊗ V −1 L ), whereM = M 0 0 0 0 . . . 0 0 0 M L−1 , and M j = [f e (α j )] e∈In(t) .

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A class of non-linearly solvable networks

Cited by 6 publications

References 22 publications

A Local Perspective on the Edge Removal Problem

A Local Perspective on the Edge Removal Problem

A class of non-linearly solvable networks

Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths

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