A New Upperbound on the Broadcast Rate and Near-Optimal Vector Linear Codes for Index Coding Problems with Symmetric Neighboring Interference

Abstract

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has equal number of K messages and K receivers, the kth receiver R k wanting the kth message x k and having the side-information. , x k+D } is the interference with D messages after and U messages before its desired message. The single unicast index coding problem with symmetric neighboring interference (SUICP-SNI) is motivated by topological interference management problems in wireless communication networks. Maleki, Cadambe and Jafar obtained the capacity of this SUICP-SNI with K tending to infinity and Blasiak, Kleinberg and Lubetzky for the special case of (D = U = 1) with K being finite. Finding the capacity of the SUICP-SNI for arbitrary K, D and U is a challenging open problem. In our previous work, for an SUICP-SNI with arbitrary K, D and U , we defined a set S of 2-tuples such that for every (a, b) in that set S, the rate D + 1 + a b is achieved by using vector linear index codes over every finite field. In this paper, we give an algorithm to find the values of a and b such that (a, b) ∈ S and a b is minimum. We present a new upperbound on the broadcast rate of SUICP-SNI and prove that this upper bound coincides with the existing results on the exact value of the capacity of SUICP-SNI in the respective settings.

Reduced Complexity Index Codes and Improved Upperbound on Broadcast Rate for Neighboring Interference Problems

“…In the b-dimensional vector linear index code construction for SUICP(SNI) given in [11], the Kb messages corresponding to K receivers are linearly combined to give b(D + 1) + a index code symbols by using an AIR matrix of size Kb × (b(D + 1) + a). In Theorem 1, we partition the Kb message symbols into b(U + 1) + c sets for some non-negative integer c such that each set comprises of Kb b(U+1)+c messages.…”

“…for i ∈ [0 : τ − 1]. We prove that receiver R k decodes its wanted messages x k,j for every k ∈ [0 : K − 1] and j ∈ [1 : b] by using γ index code symbols out of available b(D+1)+a index code symbols given in (11). Let (kb + j) mod τ = g j and kb+j τ = h j for every j ∈ [1 : b].…”

“…In A a , receiver R k knows t − γ messages as shown in (10) and (12). Hence, after subtracting the known information, (12) is a set of γ equations in γ unknowns.…”

“…In A a , receiver R k knows t − γ messages as shown in (10) and (12). Hence, after subtracting the known information, (12) is a set of γ equations in γ unknowns. The γ equations in (12) are independent follows from the fact that any γ adjacent rows in L t×γ are linearly independent.…”

“…Hence, after subtracting the known information, (12) is a set of γ equations in γ unknowns. The γ equations in (12) are independent follows from the fact that any γ adjacent rows in L t×γ are linearly independent. Hence, R k can decode its wanted message y gj +bτ = x k,j ∀j.…”