# A construction of UD $k$-ary multi-user codes from $(2^m(k-1)+1)$-ary codes for MAAC

## TL;DR

This paper presents a recursive construction method for multi-user codes in the MAAC setting, utilizing higher-ary codes to improve overall rate efficiency.

## Contribution

It introduces a novel recursive construction of UD $k$-ary multi-user codes from $(2^m(k-1)+1)$-ary codes, enhancing code rate performance.

## Key findings

- Constructed $k$-ary $T$-user UD codes from difference sets.
- Recursive method increases code rate compared to conventional codes.
- Higher-ary codes lead to improved total rate of the code set.

## Abstract

In this paper, we proposed a construction of a UD $k$-ary $T$-user coding scheme for MAAC. We first give a construction of $k$-ary $T^{f+g}$-user UD code from a $k$-ary $T^{f}$-user UD code and a $k^{\pm}$-ary $T^{g}$-user difference set with its two component sets $\mathcal{D}^{+}$ and $\mathcal{D}^{-}$ {\em a priori}. Based on the $k^{\pm}$-ary $T^{g}$-user difference set constructed from a $(2k-1)$-ary UD code, we recursively construct a UD $k$-ary $T$-user codes with code length of $2^m$ from initial multi-user codes of $k$-ary, $2(k-1)+1$-ary, \dots, $(2^m(k-1)+1)$-ary. Introducing multi-user codes with higer-ary makes the total rate of generated code $\mathcal{A}$ higher than that of conventional code.