# On Error Detection in Asymmetric Channels

## TL;DR

This paper investigates error detection in q-ary asymmetric channels with various noise restrictions, providing optimal code constructions and proving their optimality in large alphabet limits.

## Contribution

It introduces optimal error-detecting codes for asymmetric channels under multiple noise constraints and establishes their optimality in the large alphabet limit.

## Key findings

- Optimal codes are described for specific parameter sets.
- Codes are proven optimal in the large alphabet limit.
- The study covers both standard and cyclic channel models.

## Abstract

We study the error detection problem in $ q $-ary asymmetric channels wherein every input symbol $ x_i $ is mapped to an output symbol $ y_i $ satisfying $ y_i \geq x_i $. A general setting is assumed where the noise vectors are (potentially) restricted in: 1) the amplitude, $ y_i - x_i \leq a $, 2) the Hamming weight, $ \sum_{i=1}^n 1_{\{y_i \neq x_i\}} \leq h $, and 3) the total weight, $ \sum_{i=1}^n (y_i - x_i) \leq t $. Optimal codes detecting these types of errors are described for certain sets of parameters $ a, h, t $, both in the standard and in the cyclic ($ \operatorname{mod}\, q $) version of the problem. It is also demonstrated that these codes are optimal in the large alphabet limit for every $ a, h, t $ and every block-length $ n $.