Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unweighted linear congruences with distinct coordinates and the
  Varshamov--Tenengolts codes   In this paper, we first give explicit formulas for the number of solutions of
unweighted linear congruences with distinct coordinates. Our main tools are
properties of Ramanujan sums and of the discrete Fourier transform of
arithmetic functions. Then, as an application, we derive an explicit formula
for the number of codewords in the Varshamov--Tenengolts code $VT_b(n)$ with
Hamming weight $k$, that is, with exactly $k$ $1$'s. The Varshamov--Tenengolts
codes are an important class of codes that are capable of correcting asymmetric
errors on a $Z$-channel. As another application, we derive Ginzburg's formula
for the number of codewords in $VT_b(n)$, that is, $|VT_b(n)|$. We even go
further and discuss connections to several other combinatorial problems, some
of which have appeared in seemingly unrelated contexts. This provides a general
framework and gives new insight into all these problems which might lead to
further work.

13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat