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

Abstract

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has K messages and K receivers, the kth receiver R k wanting the kth message x k and having the interference with D messages after and U messages before its desired message. Maleki, Cadambe and Jafar studied SUICP(SNI) because of its importance in topological interference management problems. Maleki et. al. derived the lowerbound on the broadcast rate of this setting to be D + 1. In our earlier work, for SUICP(SNI) with arbitrary K, D and U , we defined set S of 2-tuples and for every (a, b) ∈ S, we constructed bdimensional vector linear index code with rate D + 1 + a b by using an encoding matrix of dimension Kb × (b(D + 1) + a). In this paper, we use the symmetric structure of the SUICP(SNI) to reduce the size of encoding matrix by partitioning the message symbols. The rate achieved in this paper is same as that of the existing constructions of vector linear index codes. More specifically, we construct b-dimensional vector linear index codes for SUICP(SNI) by partitioning the Kb messages into b(U +1)+c sets for some non-negative integer c. We use an encoding matrix of size Kb b(U +1)+c × b(D+1)+a b(U +1)+c to encode each partition separately. The advantage of this method is that the receivers need to store atmost b(D+1)+a b(U +1)+c number of broadcast symbols (index code symbols) to decode a given wanted message symbol. We also give a construction of scalar linear index codes for SUICP(SNI) with arbitrary K, D and U . We give an improved upperbound on the braodcast rate of SUICP(SNI).All the subscripts in SUICP(SNI) are to be considered modulo K.

Index Codes with Minimum Locality for Three Receiver Unicast Problems

“…For any two vertices i and j in G * , i, j ∈ [12], we say that there is a bidirectional edge between i and j in G * if both (i, j) and (j, i) are edges in G * . Note that there is a bidirectional edge between i and j in G * if and only if {i, j} is an edge in G * u .…”

“…For any S ⊆ [12], let G * S and G * S,u be the subgraphs of G * and G * u , respectively, induced by the vertices S. Note that G * S,u is the underlying undirected graph corresponding to G * S . Lemma 3.…”

“…locally decodable index codes is to construct coding schemes that simultaneously minimize rate and locality, and attain the optimal trade-off between these two parameters. Locally decodable index codes reduce the number of transmissions that any receiver has to listen to, and hence, reduce the receiver complexity as well, see for instance [12]. When index codes are to be used in a fading wireless broadcast channel, the probability of error at the receivers can be reduced by using index codes with small locality [13].…”

“…Prior work on locally decodable index codes have considered single unicast index coding problems [10]- [12], problems where each receiver demands exactly one message [14], and a family of index coding problems called single uniprior 1 [13], [15]. The exact characterization of the optimal trade-off between rate and locality is known in only a few cases, all of them being single unicast problems: optimal rate among nonlinear codes for single unicast problems and locality equal to one [11], rate-locality trade-off among linear codes for singleunicast problems when min-rank is one less than the number of receivers [16].…”

On Locally Decodable Index Codes

“…It is straightforward to verify that P AP P AP P AP T fits the graph σ(G). Since σ is an automorphism of G, 22 we have σ(G) = G, and hence, P AP P AP P AP T fits G. Since the column space of P L P L P L is identical to that of P AP P AP P AP T , P L P L P L represents a valid scalar linear index code for G. Extending this argument we deduce that for any j ∈ [N ], the matrix A A A (j) = P P P j A A A(P P P j ) T fits G, and the column space of L L L (j) = P P P j L L L is identical to that of A A A (j) . The matrices L L L (j) and A A A (j) , j ∈ [N ], represent N valid scalar linear codes and the corresponding fitting matrices for G. The receiver Rx i has locality 1 when using the j th code L L L (j) if the i th column of A A A (j) appears as a column of L L L (j) , else its locality is at the most .…”