The degree of a polynomial family over GF(13), enumerated

The degree of a polynomial family over GF(13), enumerated

Every member of each family is walked — 28,561 of them for the largest — and for j keys at a time the 13^j tuples of values they produce are tabulated. j-wise independence is the statement that every tuple occurs exactly the same number of times, so a cell marked exact is a fact with no tolerance in it. Beyond the degree the failure is not a bias: whole tuples are produced by no member at all, because a polynomial through k points is unique and the value at the next point is decided. The percentage in each cell is how many of the tuples are unreachable.

Drawn at 700 × 206, wide on the page. Everything above is what sign-degree returns with no arguments; the caption is the generator's own, computed from the numbers in the drawing rather than written beside it.

2 essays call sign-degree. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about — which is what optcheck and figfill exist to catch.