# Order 33

## A conjectured optimal Gram matrix

For order 33 there is a conjectured D-optimal Gram matrix G due to Bruce Solomon (18 June 2002). It is given here in compressed format, and in full here on Will Orrick's website.

The determinant of G is (441*2^74)^2.

G is not unique - there is another Gram matrix (given here in compressed format) with the same determinant but different characteristic equation. On this page we restrict our attention to solutions with Gram matrix G.

## Solomon's conjectured maxdet matrix R

The Gram matrix G may be written as G = RR^T, where the (conjectured maxdet) matrix R was found by Bruce Solomon (18 June 2002). It is given here or in full on Will Orrick's website.

By row/column switching the matrix R generates a "giant" self-dual ST-class of size 2867510353 HT-classes. In terms of Hadamard classes, the size of the giant class is 5735015027 = 2*2867510353 - 5679, since every non-self-dual HT class splits into two Hadamard classes.

## Some generators

A "generator" can be used to find several (or many) HT-classes or Hadamard classes of solutions, using the operations of row/column switching and taking duals (transposes).

The largest known class, apart from the "giant" class, is a class of size 2136 (measured in ST-classes). The full class is here and a generator is here.

Generators for the 20 largest known ST classes (excluding the giant class) are here. They generate 22888 HT-classes of (conjectured) maxdet matrices. The full set of 22888 HT-classes is here. They all have determinant +-441*2^74.

105485 HT classes not in the giant class are here (16MB, compressed using bzip2). Row/column switching (and transposition) from this list can generate a total of 3564048 HT-equivalence classes (7126516 Hadamard equivalence classes) not in the giant class. This requires 63 iterations. The complete file of 3564048 HT-equivalence classes is here (453MB, compressed using bzip2).

## Graphs of some ST classes

Here are the graphs of some ST classes:

ST10a..ST19a (one each of sizes 10..19)
ST66a (size 66)
ST100a (size 100)
ST200a (size 200)
ST999a (size 999)
ST999b (size 999, different)
ST2136a (size 2136)

## Singletons and self-duals

We know many singletons (matrices that give an ST class of size 1) and many self-dual matrices (matrices H-equivalent to their duals), as well as 392 matrices which have both properties. An example is given here. A list of 5183 known singletons (including 260 selfdual) is here, and a list of all (up to equivalence) 5679 self-dual matrices in the giant class is here. A list of 1320 known self-dual matrices that are not singletons but are not in the giant class, so generate small but nontrivial ST classes, is here.