Lucas's theorem and some related results for extended pascal triangles
American Mathematical Monthly
Representing Boolean functions as polynomials modulo composite numbers
Computational Complexity - Special issue on circuit complexity
IEEE/ACM Transactions on Networking (TON) - Special issue on networking and information theory
Index Coding with Side Information
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Broadcasting with Side Information
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Linear index coding via semidefinite programming
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
| Hi-index | 754.84 |
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,..., Rn. The receiver Ri is interested in xi, 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 Ri knows the bit xj. An index code for G is an encoding scheme which enables each Ri to always reconstruct xi, given his side information. The minimal word length of an index code was studied by Bar-Vossef, Birk, Jayram, and Kol (FOCS'06). They introduced a graph parameter, minrk2(G), which completely characterizes the length of an optimal linear index code for G. They 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 Bar-Vossef, Birk, Jayram, and Kol in the following strong sense: for any ∈ 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 n1-∈, and yet a nonlinear index code for G has a word length of n∈. 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).