The Probability of Undetected Error for Binary Constant-Weight Codes

In this correspondence, we study the probability of undetected error for binary constant-weight codes. First, we derive a new formula on the probability of undetected error for binary constant-weight codes. Second, using this new formula and linear programming, we give two new lower bounds on the probability of undetected error for binary constant-weight codes. These two new lower bounds improve on previously known lower bounds in certain cases. Furthermore, we show that these two lower bounds are tight if and only if the binary constant-weight codes are generated from certain -designs in combinatorial design theory. This means that these binary constant-weight codes generated from certain -designs are uniformly optimal for error detection. Along the way, we determine the distance distributions of such binary constant-weight codes. Finally, several examples are given to illustrate the results obtained in this correspondence.