  The capacity of symmetric, neighboring and consecutive side-information
single unicast index coding problems (SNC-SUICP) with number of messages equal
to the number of receivers was given by Maleki, Cadambe and Jafar. For these
index coding problems, an optimal index code construction by using Vandermonde
matrices was proposed. This construction requires all the side-information at
the receivers to decode their wanted messages and also requires large field
size. In an earlier work, we constructed binary matrices of size $m \times n
(m\geq n)$ such that any $n$ adjacent rows of the matrix are linearly
independent over every field. Calling these matrices as Adjacent Independent
Row (AIR) matrices using which we gave an optimal scalar linear index code for
the one-sided SNC-SUICP for any given number of messages and one-sided
side-information. By using Vandermonde matrices or AIR matrices, every receiver
needs to solve $K-D$ equations with $K-D$ unknowns to obtain its wanted
message, where $K$ is the number of messages and $D$ is the size of the
side-information. In this paper, we analyze some of the combinatorial
properties of the AIR matrices. By using these properties, we present a
low-complexity decoding which helps to identify a reduced set of
side-information for each users with which the decoding can be carried out. By
this method every receiver is able to decode its wanted message symbol by
simply adding some index code symbols (broadcast symbols). We explicitly give
both the reduced set of side-information and the broadcast messages to be used
by each receiver to decode its wanted message. For a given pair or receivers
our decoding identifies which one will perform better than the other when the
broadcast channel is noisy.
