论文标题
未加权的线性一致性与不同的坐标和varshamov- -tenengolts代码
Unweighted linear congruences with distinct coordinates and the Varshamov--Tenengolts codes
论文作者
论文摘要
在本文中，我们首先给出明确的公式，以与不同坐标的未加权线性一致性解决方案数量。我们的主要工具是Ramanujan总和的属性和算术函数的离散傅立叶变换的属性。然后，作为一个应用程序，我们为Varshamov--tenengolts代码$ vt_b（n）$带有hamming strize $ k $的代码中的代码字数的明确公式，即$ k $ $ 1 $ $。 Varshamov- Tenengolts代码是一类重要的代码类，能够在$ z $ channel上纠正不对称错误。作为另一个应用程序，我们得出了Ginzburg在$ vt_b（n）$中的代码字数的公式，即$ | vt_b（n）| $。我们甚至走得更远，讨论与其他几个组合问题的联系，其中一些出现在看似无关的上下文中。这提供了一个一般框架，并对所有这些问题提供了新的见解，这可能会导致进一步的工作。
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.
