## Known bounds on maximal determinants

### N=4j

The general bound on the determinant of {-1, +1} matrices, given by Hadamard
[Had], is N^{N/2}, where N is the
order of the matrix. When N is divisible by 4, this bound is
sharp by which we mean
that there are arbitrarily high values of N that achieve it
(Sylvester matrices are one
infinite family). The Hadamard bound is achieved only for
Hadamard matrices. Paley has
conjectured that a Hadamard matrix exists for every order divisible by 4, and
thus that the bound is achievable for every such order. The lowest order for
which a
Hadamard matrix is not yet known to exist is
N = 668 [SY]. (On 21 June 2004
Hadi Kharaghani and Behruz Tayfeh-Rezaie announced the
discovery of a Hadamard matrix of order
428. Since 1985, this had been the lowest outstanding
order [Sa].)
Except for N=1, 2, the Hadamard bound is never achievable when N is not a
multiple of 4. Better bounds have been given according to whether N is
congruent to 1, 2 or 3 modulo 4, as set out below.

### N=4j+1

Barba gave the bound (2N-1)^{(1/2)} (N-1)^{(N-1)/2}
[Ba]. It was rederived by Ehlich
[E1] and Wojtas
[Wo].
(It actually holds for all odd cases, but a better one was found by
Ehlich for N=4j+3 [E2].) Clearly the bound
can only be achievable when
2N-1 is a perfect square. The bound is sharp, in the sense defined above,
due to a construction of Brouwer [Br].
Writing N=2q^{2}+2q+1, his construction gives a matrix of
maximal determinant whenever q is a power of an odd prime.
The bound is also known to be achievable when N=5
[M], 13 [R]
or 41 [BHH, vT].
### N=4j+2

Ehlich [E1] and Wojtas
[Wo] independently derived the bound
2(N-1) (N-2)^{(N-2)/2}. The bound is achievable only if N-1 is the
sum of two squares. By applying the Sylvester construction to a matrix
that achieves the bound of the previous paragraph, a matrix is obtained which
achieves the Ehlich/Wojtas bound. In particular, the bound is achieved
whenever N=2(2q^{2}+2q+1) for q a power of an odd prime.
Therefore the bound is sharp. A second infinite family of constructions,
due to Koukouvinos, Kounias, Seberry, Singer and Spence
[Sp1, KKS]
is known for N=2(q^{2}+q+1) where q is a power of a prime
(even or odd).
The bound has also been achieved for certain low orders which do not fall
into either of the infinite families. Constructions
were given by Ehlich [E1] for N up to 38. Yang
[Y1-Y5] added constructions up to N=66, and
Cohn [C1, C2] extended this as far as N=102.
The case N=86 had previously been handled
by Chadjipantelis and Kounias [CK].
Furthermore, maximal matrices were found for N=110 by Fletcher and Seberry
[FS], for N=118 by Fletcher, Koukouvinos and
Seberry [FKS], for N=126, 186 and N=158,
194, 290 by Djokovic [D1, D2], for N=170 by
Gysin [Gys], for N=150 by
Holzmann and Kharaghani [HK], and for N=206,
242, 262, 482 by Djokovic and Kotsireas [DK].
The lowest outstanding order is 138.

### N=4j+3

Ehlich [E2] derived the bound

(n-3)^{(n-s)/2} (n-3+4r)^{u/2} (n+1+4r)^{v/2}
{1 - ur / (n-3+4r) - v(r+1) / (n+1+4r)}^{1/2}

where s=3 for n=3, s=5 for n=7, s=5 or 6 for n=11, s=6 for n=15, 19, ..., 59,
and s=7 for n ≥ 63, r = [n/s], n = rs + v and u = s - v. Here [n/s]
represents the floor of n/s. Cohn [C4]
showed that the bound is an integer
only when n=112t^{2} ± 28t + 7 for some positive integer t.
It is not known if the bound is sharp with the meaning used above, or even
if it is achievable beyond n=3.

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Page created 2 March 2002.

Last modified 3 May 2007.

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