On Error Detection in Asymmetric Channels

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 $n$.