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- W4289655110 abstract "It has been known that the insufficiency of linear coding in achieving the optimal rate of the general index coding problem is rooted in its rate’s dependency on the field size. However, this dependency has been described only through the two well-known matroid instances, namely the Fano and non- Fano matroids, which, in turn, limits its scope only to the fields with characteristic two. In this paper, we extend this scope to demonstrate the reliance of linear coding rate on fields with characteristic three. By constructing two index coding instances of size 29, we prove that for the first instance, linear coding is optimal only over the fields with characteristic three, and for the second instance, linear coding over any field with characteristic three can never be optimal. Another main contribution of this paper is to reduce the key constraints on the space of the linear coding for each index coding instance of size 29 into a matroid instance with the ground set of size 9, whose linear representability is dependent on the fields with characteristic three. The proofs and discussions provided in this paper through using these two relatively small matroid instances will shed light on the underlying reason causing the linear coding to become insufficient for the general index coding problem." @default.
- W4289655110 created "2022-08-03" @default.
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- W4289655110 date "2022-06-26" @default.
- W4289655110 modified "2023-10-16" @default.
- W4289655110 title "On the Optimality of Linear Index Coding over the Fields with Characteristic Three" @default.
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- W4289655110 doi "https://doi.org/10.1109/isit50566.2022.9834753" @default.
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