### 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 2^{N-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.

Comments:
maxdet@indiana.edu