In this work, we consider $q$-ary signature codes of length $k$ and size $n$ for a noisy adder multiple access channel. A signature code in this model has the property that any subset of codewords can be uniquely reconstructed based on any vector that is obtained from the sum (over integers) of these codewords. We show that there exists an algorithm to construct a signature code of length $k = \frac{2n\log{3}}{(1-2\tau)\left(\log{n} + (q-1)\log{\frac{\pi}{2}}\right)} +\mathcal{O}\left(\frac{n}{\log{n}(q+\log{n})}\right)$ capable of correcting $\tau k$ errors at the channel output, where $0\le \tau < \frac{q-1}{2q}$. Furthermore, we present an explicit construction of signature codewords with polynomial complexity being able to correct up to $\left( \frac{q-1}{8q} - \epsilon\right)k$ errors for a codeword length $k = \mathcal{O} \left ( \frac{n}{\log \log n} \right )$, where $\epsilon$ is a small non-negative number. Moreover, we prove several non-existence results (converse bounds) for $q$-ary signature codes enabling error correction.

翻译：在这项工作中, 我们考虑使用 $q$- y 签名代码的长度 $k 和大小 $n 美元 。 本模型中的签名代码具有这样的属性, 任何组代号都可以根据从这些代号的总和( 超整数) 中获得的任何矢量进行独特的重建。 我们显示, 存在一种算法来构建一个长度的签名代码 $k =\ frac{ q\\ log{ 3} (1-2\ tau)\ left( log{ right} + (q-1\ log_ right) + (q-1\ log\\\ right) + (q-1\ log\\ right)\\\\\\ left( leftc) comm comm codeal 能够修正到 $left $ nc_ rick_ rick_ $\ rick_\\\\\ rick\ rick\ rick_ reck\ reck reck reck reck reck reck_ reck reck reck reck a dex) 。 我们提出了 =0\ = = = qq_ qq_ = = reck = = a = a reck = = = a rick reck = = = =xxxxxxxxxxxxxx= = = = = = = = = = = = = = = = = = =xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx