**2**Author(s)

# On Network Coding for Sum-Networks

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A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear net- work coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under nonlinear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.

- Page(s):
- 50 - 63
- ISSN :
- 0018-9448
- INSPEC Accession Number:
- 12443717
- DOI:
- 10.1109/TIT.2011.2169532

- Date of Current Version :
- 05 January 2012
- Issue Date :
- Jan. 2012
- Sponsored by :
- IEEE Information Theory Society
- Publisher:
- IEEE