Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: New Results on Two Hypercube Coloring Problems   In this paper, we study the following two hypercube coloring problems: Given
$n$ and $d$, find the minimum number of colors, denoted as ${\chi}'_{d}(n)$
(resp. ${\chi}_{d}(n)$), needed to color the vertices of the $n$-cube such that
any two vertices with Hamming distance at most $d$ (resp. exactly $d$) have
different colors. These problems originally arose in the study of the
scalability of optical networks. Using methods in coding theory, we show that
${\chi}'_{4}(2^{r+1}-1)=2^{2r+1}$, ${\chi}'_{5}(2^{r+1})=4^{r+1}$ for any odd
number $r\geq3$, and give two upper bounds on ${\chi}_{d}(n)$. The first upper
bound improves on that of Kim, Du and Pardalos. The second upper bound improves
on the first one for small $n$. Furthermore, we derive an inequality on
${\chi}_{d}(n)$ and ${\chi}'_{d}(n)$.

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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph