Figure 1

Guaranteed error correction and detection properties of STD. An experimenter, expecting up to t positives and E errors, chooses a satisfactory prime number q and builds the set of pools STD(n; q; t·Γ+2·E+1), as specified in corollary 2. Recall that n is the total number of variables and Γ is the compression power, i.e. the smallest γ such that q^γ+1 ≥ n. This figure summarizes the behavior of these pools when the actual number of errors exceeds E, and distinguishes between the two types of errors: false positives and false negatives. In the dark blue region, all errors are detected and corrected. In the intermediate blue rectangles, correction is not guaranteed but detection is: in an unfavorable conformation of positives and errors, correction of all errors may fail, but this failure cannot go unnoticed, and the user can therefore plan additional experiments. In the cyan square, detection is usually also guaranteed, except if E is very small (E < 2·Γ-1): in this case, the line y = 3·E+1-x splits the square in two, and detection is only guaranteed in the bottom left portion, where the total number of errors is at most 3·E+1. Finally, in the outer pale cyan zone, no guarantee is provided.