  In this paper, linear index codes with multiple senders are studied, where
every receiver receives encoded messages from all senders. A new fitting matrix
for the multiple senders is proposed and it is proved that the minimum rank of
the proposed fitting matrices is the optimal codelength of linear index codes
for the multiple senders. In addition, a new type of a side information graph
related with the optimal codelength is proposed and whether given side
information is critical or not is studied. Furthermore, linear index codes for
the cellular network scenario are studied, where each receiver can receive a
subset of sub-codewords. Since some receivers cannot receive the entire
codeword in the cellular network scenario, the encoding method based on the
fitting matrix has to be modified. In the cellular network scenario, we propose
another fitting matrix and prove that an optimal generator matrix can be found
based on these fitting matrices. In addition, some properties on the optimal
codelength of linear index codes for the cellular network case are studied.
