Index Coding: Rank-Invariant Extensions

An index coding (IC) problem consisting of a server and multiple receivers with different side-information and demand sets can be equivalently represented using a fitting matrix. A scalar linear index code to a given IC problem is a matrix representing the transmitted linear combinations of the message symbols. The length of an index code is then the number of transmissions (or equivalently, the number of rows in the index code). An IC problem ${\cal I}{ext}$iscalledanextensionofanotherICproblem${\cal I}$ if the fitting matrix of $\mathcal{I}$ is a submatrix of the fitting matrix of ${\cal I}{ext}$\mathrm{.}Wefirstpresentastraightforward$m$\mathit{\text{-order}}extension${\cal I}_{ext}$ofanICproblem${\cal I}$ for which an index code is obtained by concatenating $m$ copies of an index code of $\mathcal{I}$. The length of the codes is the same for both $\mathcal{I}$ and ${\mathcal{I}}_{ext}$, and if the index code for $\mathcal{I}$ has optimal length then so does the extended code for ${\mathcal{I}}_{ext}$. More generally, an extended IC problem of $\mathcal{I}$ having the same optimal length as $\mathcal{I}$ is said to be a $\mathit{\text{rank-invariant}}$ extension of $\mathcal{I}$. We then focus on $2$-order rank-invariant extensions of $\mathcal{I}$, and present constructions of such extensions based on involutory permutation matrices.