Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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 index 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. Then, a variation of the second
instance is designed as the third index coding instance of size 58. For this
instance, it is proved that while linear coding over any field with
characteristic three cannot be optimal, there exists a nonlinear code over the
fields with characteristic three, which achieves its optimal rate. Connecting
the first and third index coding instances in two specific ways, called no-way
and two-way connections, will lead to two new index coding instances of size 87
and 91, for which linear coding is outperformed by nonlinear codes.
  Another main contribution of this paper is the reduction of the key
constraints on the space of the linear coding for the first and second index
coding instances, each 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.

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The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
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The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys