There is a mod 4 dependence, hence the four columns. The entries in the first column all achieve the Hadamard bound and the regular pattern observed for that column will continue indefinitely if the Hadamard conjecture is correct. Since the discovery of a Hadamard matrix of order 428 by H. Kharaghani and B. Tayfeh-Rezaie, the lowest order for which a Hadamard matrix is not yet known to exist is N = 668 [SY]. There are also proved general bounds for the other three columns [Ba, E1, E2, Wo].
The factorization of the determinants generally obeys the following rule: for row j+1 of the table, the largest power of j not exceeding 2j-1 has been extracted.
Determinants displayed in white are known to be the biggest possible. Entries tagged with ?? have not been proved maximal.