Signature Codes for a Noisy Adder Multiple Access Channel

TL;DR

This paper introduces new constructions of $q$-ary signature codes for noisy adder multiple access channels, enabling efficient error correction and providing bounds on their existence.

Contribution

It presents an algorithm for constructing signature codes with error correction capabilities and explicit polynomial-time constructions for certain parameters, along with non-existence bounds.

Findings

Existence of algorithms for constructing error-correcting signature codes.

Explicit polynomial-time constructions for certain code parameters.

Non-existence bounds for $q$-ary signature codes with error correction.

Abstract

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.