Sufficient Conditions for Monotonicity of the Undetected Error Probability for Large Channel Error Probabilities

The performance of a linear error-detecting code in a symmetric memoryless channel is characterized by its probability of undetected error, which is a function of the channel symbol error probability, involving basic parameters of a code and its weight distribution. However, the code weight distribution is known for relatively few codes since its computation is an NP-hard problem. It should therefore be useful to have criteria for properness and goodness in error detection that do not involve the code weight distribution. In this work we give two such criteria. We show that a binary linear code $C$ of length $n$ and its dual code $C^\perp$ of minimum code distance $d^\perp$ are proper for error detection whenever $d^\perp\geqslant\lfloor n/2\rfloor+1$, and that $C$ is proper in the interval $[(n+1-2d^\perp)/(n-d^\perp),1/2]$ whenever $\lceil n/3\rceil+1\leqslant d^\perp\leqslant\lfloor n/2\rfloor$. We also provide examples, mostly of Griesmer codes and their duals, that satisfy the above conditions.