Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Constructions of Optimal Cyclic $(r,\delta)$ Locally Repairable Codes   A code is said to be a $r$-local locally repairable code (LRC) if each of its
coordinates can be repaired by accessing at most $r$ other coordinates. When
some of the $r$ coordinates are also erased, the $r$-local LRC can not
accomplish the local repair, which leads to the concept of
$(r,\delta)$-locality. A $q$-ary $[n, k]$ linear code $\cC$ is said to have
$(r, \delta)$-locality ($\delta\ge 2$) if for each coordinate $i$, there exists
a punctured subcode of $\cC$ with support containing $i$, whose length is at
most $r + \delta - 1$, and whose minimum distance is at least $\delta$. The
$(r, \delta)$-LRC can tolerate $\delta-1$ erasures in total, which degenerates
to a $r$-local LRC when $\delta=2$. A $q$-ary $(r,\delta)$ LRC is called
optimal if it meets the Singleton-like bound for $(r,\delta)$-LRCs. A class of
optimal $q$-ary cyclic $r$-local LRCs with lengths $n\mid q-1$ were constructed
by Tamo, Barg, Goparaju and Calderbank based on the $q$-ary Reed-Solomon codes.
In this paper, we construct a class of optimal $q$-ary cyclic $(r,\delta)$-LRCs
($\delta\ge 2$) with length $n\mid q-1$, which generalizes the results of Tamo
\emph{et al.} Moreover, we construct a new class of optimal $q$-ary cyclic
$r$-local LRCs with lengths $n\mid q+1$ and a new class of optimal $q$-ary
cyclic $(r,\delta)$-LRCs ($\delta\ge 2$) with lengths $n\mid q+1$. The
constructed optimal LRCs with length $n=q+1$ have the best-known length $q+1$
for the given finite field with size $q$ when the minimum distance is larger
than $4$.

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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org