Verification of Conjecture 6.1 for all fields with less than 10^13 elements

The data file lists the various field sizes p^n (p prime, n a positive integer) that it checked.  It skips over n that are multiples of 2 or 3, for Conjecture 6.1 is verified by Theorem 1.1 when n is a power of 2 or a multiple of 3.  When n is even but not a power of 2, either d is degenerate in the largest proper subfield (and then Conjecture 6.1 is verified by Corollary 4.4 of Aubry, Katz, Langevin, "Cyclotomy of Weil Sums of Binomials", Journal of Number Theory 154, 160-178 (2015)).  Or else d is nondegenerate over that proper subfield, in which case it is assumed that Conjecture 6.1 has already been verified in this smaller field, and thus by lifting the solution via Lemma 2.7 of the paper, the conjecture is also verified for the current field.

For each prime power not already set aside, the program runs through all possible exponents d in the Weil sum, proceeding in order of p-ary weight.  The program only checks those exponents d of sufficiently low p-ary weight that the conjecture would not be immediately verified by setting a=1 in the first expression in Lemma 2.9.   And the program skips d that are not congruent to 1 modulo p-1, for Conjecture 6.1 is automatically verified for these by Theorem 1.1(iii). 

Among those d with the same p-ary weight, the program proceeds in numerical order.  For any integer k, p^k*d reduced modulo p^n-1 (a cyclic shift of d modulo p^n-1 in p-ary notation) gives the same spectrum of Weil sum values as d.  So we shall later apply a filter that skips d (as already having been checked) if d is not the least such cyclic shift.  We do not go to any value of d so high that it is obvious that some cyclic shift of it modulo p^n-1 will be lower than d itself.

Since d^(-1) (mod p^n-1) and its cyclic shifts also give the same spectrum of Weil sums as d, then we shall later apply a filter that skips d (as having already been checked) if d^(-1) has lower p-ary weight that d, or equal p-ary weight but lower numerical value.

Once we restrict to such d, there are various filters employed which filter out all d that either do not meet the hypotheses of the conjecture or for which the program can verify the conjecture.  The program lists how many of the d values checked are filtered out by a particular filter, recording the number of d not filtered by any filter as the number of "losses."  For the fields checked, we have no losses, thus verifying the conjecture for all fields under consideration.

We now describe the specific filters:

The filter "niho" checks if d is not a prime power modulo p^m-1 for some m < n, in which case it is assumed that the conjecture has already been verified in this smaller field, and thus (by lifting the solution via Lemma 2.7 of the paper) also verified for the current field, and hence filtered out.

The filter "shift" checks if there is a k such that p^k*d (mod p^n-1) (which has the same spectrum of Weil sum values as d) is lower than d, and hence has already been checked, so that d can be filtered out.

The filter "coprime" checks if d is not coprime to p^n-1, and hence not under consideration in Conjecture 6.1, and thus filtered out.

The filter "inv wt" checks if p-ary weight of d^(-1) (which has the same spectrum of Weil sum values as d) is larger than that of d, and hence has already been checked, so that d can be filtered out. 

If the p-ary weight of d^(-1) is equal to that of d, the filter "shift" checks if there is a k such that p^k*d^(-1) (mod p^n-1) (which has the same spectrum of Weil sum values as d) is lower than d, and hence has already been checked, so that d can be filtered out.

The filter "hard" exhaustively checks for an a in Lemma 2.9 of the paper that would verify the conjecture for d, so that d can be filtered out.