On the Optimality of Linear Index Coding Over the Fields With Characteristic Three

It has been known that the insufficiency of linear coding in achieving optimal rate general index problem is rooted its rate’s dependency on field size. However, this described only through two well-known matroid instances, namely Fano and non-Fano matroids, which, turn, limits scope to fields with characteristic two. In paper, we extend demonstrate reliance three. By constructing instances size 29, prove for first instance, over three, second any three can never be optimal. Then, a variation instance designed as third 58. For it proved while cannot optimal, there exists nonlinear code which achieves rate. Connecting specific ways, called no-way two-way connections, will lead new 87 91, outperformed by codes. Another main contribution paper reduction key constraints space each into ground set 9, whose representability dependent The proofs discussions provided using these relatively small shed light underlying reason causing become insufficient problem.