A simple way to compute minimum Euclidean distance for synchronous coded multiuser systems

We show that the minimum squared Euclidean distance for a synchronous multiuser system using convolutional codes is no less than the product of the free distance of the code and the minimum Euclidean distance for the corresponding uncoded synchronous multiuser system. When all users use an identical convolutional code, equality holds. Thus the relationship can be used to compute the minimum squared Euclidean distance for a coded system. The results also indicate that in terms of maximizing the minimum squared Euclidean distance, it is better if all users use nonidentical error control codes.