  An index code is said to be locally decodable if each receiver can decode its
demand using its side information and by querying only a subset of the
transmitted codeword symbols instead of observing the entire codeword. Local
decodability can be a beneficial feature in some communication scenarios, such
as when the receivers can afford to listen to only a part of the transmissions
because of limited availability of power. The locality of an index code is the
ratio of the maximum number of codeword symbols queried by a receiver to the
message length. In this paper we analyze the optimum locality of linear codes
for the family of index coding problems whose min-rank is one less than the
number of receivers in the network. We first derive the optimal trade-off
between the index coding rate and locality with vector linear coding when the
side information graph is a directed cycle. We then provide the optimal
trade-off achieved by scalar linear coding for a larger family of problems,
viz., problems where the min-rank is only one less than the number of
receivers. While the arguments used for achievability are based on known coding
techniques, the converse arguments rely on new results on the structure of
locally decodable index codes.
