Capacity of Some Index Coding Problems with Symmetric Neighboring Interference

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has equal number of K messages and K receivers, the kth receiver Rk wanting the kth message xk and having the side-information Kk = (Ik∪xk) , where Ik = {xk−U , . . . , xk−2, xk−1}∪{xk+1, xk+2, . . . , xk+D} is the interference with D messages after and U messages before its desired message. Maleki, Cadambe and Jafar obtained the capacity of this symmetric neighboring interference single unicast index coding problem (SNI-SUICP) with (K) tending to infinity and Blasiak, Kleinberg and Lubetzky for the special case of (D = U = 1) with K being finite. In this work, for any finite K and arbitrary D we obtain the capacity for the case U = gcd(K,D+1)−1. Our proof is constructive, i.e., we give an explicit construction of a linear index code achieving the capacity.