The capacity of symmetric instance of the multiple unicast index coding problem with neighboring antidotes (sideinformation) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar in [1]. In this paper we consider ten symmetric multiple unicast problems with lesser antidotes than considered in [1] and explicitly construct scalar linear codes for them. These codes are shown to achieve the capacity or equivalently these codes shown to be of optimal length. Also, the constructed codes enable the receivers use small number of transmissions to decode their wanted messages which is important to have the probability of message error reduced in a noisy broadcast channel [8], [10]. Some of the cases considered are shown to be critical index coding problems and these codes help to identify some of the subclasses considered in [1] to be not critical index coding problems. 1 min(U,D)+1 K+min(U,D)−max(U,D)

A single unicast index coding problem (SUICP) is called symmetric neighboring and consecutive (SNC) sideinformation problem if it has K messages and K receivers, the kth receiver R k wanting the kth message x k and having the side-information D messages immediately after x k and U (D ≥ U ) messages immediately before x k . Maleki, Cadambe and Jafar obtained the capacity of this SUICP(SNC) and proposed (U + 1)-dimensional optimal length vector linear index codes by using Vandermonde matrices. However, for a b-dimensional vector linear index code, the transmitter needs to wait for b realizations of each message and hence the latency introduced at the transmitter is proportional to b. For any given single unicast index coding problem (SUICP) with the side-information graph G, MAIS(G) is used to give a lowerbound on the broadcast rate of the ICP. In this paper, we derive MAIS(G) of SUICP(SNC) with side-information graph G. We construct scalar linear index codes for SUICP(SNC) with length K U +1 − D−U U +1 . We derive the minrank(G) of SUICP(SNC) with side-information graph G and show that the constructed scalar linear index codes are of optimal length for SUICP(SNC) with some combinations of K, D and U . For SUICP(SNC) with arbitrary K, D and U , we show that the length of constructed scalar linear index codes are atmost two index code symbols per message symbol more than the broadcast rate. The given results for SUICP(SNC) are of practical importance due to its relation with topological interference management problem in wireless communication networks.

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.

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. 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.• If L(j, k) is present in I D+1 , then j R = j and k R = k.

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