I want to make a DoE including several labs that will contribute a different number of tests to the study. I use a table (a simplified example below) so define the number of tests per lab by means of a covariate table.

For the purpose of this discussion I have use a simple model with only main effects:

This leads to a design with the alias matrix and the color map on correlations reproduced below:

The question relates to the variable "Lab". My understanding is that for all but one level of a categorical factor a new factor is created which can have the values 0 or 1. There is one variable less than the number of levels of the factor, because the sum of all of these factors levels is only 1. In the current example there are 3 Labs (A,B,C) so two new variables Lab 1 and Lab 2 are created.

First question: in the table and in the color map interaction terms like Lab*X2 1 appear instead of Lab1*X2. Why is that? Is that a kind of error somewhere in the program, or does this have a deeper reason I'm not aware of?

Second question: How do the factors Lab 1 and Lab 2 relate to the labs A,B,C. For instance if Lab C would make a systematic error but Labs A and B would work correctly and achieve the same test outcome, how would that be reflected in the model?