# Optimal $q$-Ary Error Correcting/All Unidirectional Error Detecting Codes

## TL;DR

This paper introduces combinatorial constructions for optimal q-ary codes that correct symmetric errors and detect unidirectional errors, determining exact code lengths for various parameters.

## Contribution

It presents new combinatorial methods using graph factorizations and code concatenations to construct optimal error-correcting/detecting codes, expanding known code length results.

## Key findings

- Exact values of shortest code lengths for new parameter families
- New combinatorial constructions for q-ary t-EC-AUED codes
- Extended the range of known optimal code parameters

## Abstract

Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$-EC-AUED codes, are studied in this paper. Given positive integers $q$, $a$ and $t$, let $n_q(a,t+1)$ denote the length of the shortest $q$-ary $t$-EC-AUED code of size $a$. We introduce combinatorial constructions for $q$-ary $t$-EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of $n_q(a,t+1)$ for several new infinite families of $q,a$ and $t$.