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Linear index coding via graph homomorphism

Ebrahimi, J. B ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1109/CoDIT.2014.6996886
  3. Abstract:
  4. In [1], [2] it is shown that the minimum broadcast rate of a linear index code over a finite field Fq is equal to an algebraic invariant of the underlying digraph, called minrankq. In [3], it is proved that for F2 and any positive integer k, minrankq(G) ≤ k if and only if there exists a homomorphism from the complement of the graph G to the complement of a particular undirected graph family called 'graph family {Gk}'. As observed in [2], by combining these two results one can relate the linear index coding problem of undirected graphs to the graph homomorphism problem. In [4], a direct connection between linear index coding problem and graph homomorphism problem is introduced. In contrast to the former approach, the direct connection holds for digraphs as well and applies to any field size. More precisely, in [4], a graph family {Hk q} has been introduced and shown that whether or not the scalar linear index of a digraph G is less than or equal to k is equivalent to the existence of a graph homomorphism from the complement of G to the complement of Hk q
  5. Keywords:
  6. Computational complexity of the minrank ; Linear index coding ; Directed graphs ; Graph theory ; Field size ; Finite fields ; Graph G ; Graph homomorphisms ; Index coding ; Minrank of a graph ; Positive integers ; Undirected graph ; Codes (symbols)
  7. Source: Proceedings - 2014 International Conference on Control, Decision and Information Technologies, CoDIT 2014 ; 2014 , pp. 158-163 ; ISBN: 9781479967735
  8. URL: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6996886