Nonlinear Index Coding Outperforming the Linear Optimum

Abstract

The following source coding problem was introduced by Birk and Kol: a sender holds a word x ∈ {0, 1} n , and wishes to broadcast a codeword to n receivers, R 1 , . . . , R n . The receiver R i is interested in x i , and has prior side information comprising some subset of the n bits. This corresponds to a directed graph G on n vertices, where ij is an edge iff R i knows the bit x j . An index code for G is an encoding scheme which enables each R i to always reconstruct x i , given his side information. The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram and Kol [4]. They introduced a graph parameter, minrk 2 (G), which completely characterizes the length of an optimal linear index code for G. The authors of [4] showed that in various cases linear codes attain the optimal word length, and conjectured that linear index coding is in fact always optimal.In this work, we disprove the main conjecture of [4] in the following strong sense: for any ε > 0 and sufficiently large n, there is an n-vertex graph G so that every linear index code for G requires codewords of length at least n 1−ε , and yet a non-linear index code for G has a word length of n ε . This is achieved by an explicit construction, which extends Alon's variant of the celebrated Ramsey construction of Frankl and Wilson.In addition, we study optimal index codes in various, less restricted, natural models, and prove several related properties of the graph parameter minrk(G).

Some new results on index coding when the number of data is less than the number of receivers

“…Hereafter, we shall use P i (i = 1, 2) to represent a particular index coding problem belonging to the case C i .) Recently, Lubetzky and Stav considered the case C 2 and obtained a lower bound on the minimum length of index coding for P 2 by considering a P 1 properly constructed from the original index coding problem P 2 [6].…”

“…Here, G cl (P 2 ) is a directed graph properly constructed from P 2 as in [6], and P 1 (G) is the C 1 -index coding problem equivalent to a directed graph G. (See [1] for the equivalence between G and P 1 (G).) (·) is the minimum index code length of the corresponding problem.…”

“…• Third, based on the existing symmetric capacity result for [6], and our new result, we compute the symmetric capacity for C 2 with n < m ≤ 5. In the case to which the first result in the above applies, the above result simply yields …”

“…In addition, the authors identified certain cases in which linear index coding is optimal, and such cases include directed acyclic graphs, perfect graphs, odd cycles and odd anti-holes, etc. However, in [6], Lubetzky and Stav showed that there exist certain cases in which nonlinear index coding strictly outperforms linear index coding. In [4], Arbabjolfaei et al obtained the capacity region for index coding with up to five receivers for C 1 .…”

“…For example, a C 2 -index coding problem P 2 cannot be represented by a directed graph with equivalence. However, it is shown that it is convenient to express a P 2 as a matrix in a similar way to the case C 1 [6]. Define a matrix set M (P 2 ) for P 2 as follows:…”