Non-linear index coding outperforming the linear optimum

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, R₁, . . . , R_n. The receiver R₁ 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, minrk₂(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.