Key to the table

N is the order of the matrix. Det is the largest determinant known for that order. More precisely, the determinant of an N×N sign ({-1, 1}) matrix has the stated maximum multiplied by 2N-1 (this factor is common to all sign matrices); the determinant of an (N-1)×(N-1) binary ({0, 1}) matrix has the stated maximum with no multiplier. This is a consequence of a mapping between binary and sign matrices. R is the ratio of the maximum to the general bound for the particular column in which the value appears.

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.

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Page created 25 February 2003.
Last modified 23 June 2004.