OD_Designs

A collection of previously unknown Orthogonal Designs as listed in the appendices of "Orthogonal Designs Hadamard Matrices, Quadratic Forms and Algebras" by Jennifer Seberry.

From the introduction of the above book:

An orthogonal design of order $n$, type $(s_1, \dots, s_l)$, denoted, OD(n; $s_1, \ldots, s_l)$, $s_i$ positive integers,is an $n \times n$ matrix with entries from ${0, \pm x_1, \dots, \pm x_l}$ (the $x_i$ commuting indeterminates) satisfying:

$XX^T=(\sum_{i=1}^l s_i x_i^2) I_n$

Each json file encodes an orthogonal design. For example, od24_1_1_1_1_2_5_5_8.json encodes an $OD(24;1,1,1,1,2,5,5,8)$ with $1$ replacing $x_1$ ($-1$ replacing $-x_1$), $2$ replacing $x_2$ and so on.

These designs were all produced via searches using pl_search_cpp with various techniques used to reduce the search space. For ODs with at least three singleton indeterminates the code was checked on known order 24 designs with 8 indeterminates with at least 3 being singletons. The code found solutions for all of the known designs. The listed designs where the only ones found from the unknows list, suggesting these were the only ones.

Solutions for the designs $OD(24;1,1,1,1,1,1,1,9)$, $OD(24;1,1,1,1,1,1,2,8)$ and $OD(24;1,1,1,1,1,1,5,5)$ were previously found using different techniques by Andrew Souter.

The following was generated by OD2tex.py from od24_1_1_1_1_2_5_5_8.json. For help running the program try python OD2tex.py --help.

The following alternative representation of ODs is also presented in the introduction of the book.

We can write OD(n; $s_1, \ldots, s_l)$ as $A_1x_1+\ldots+A_lx_l$ where $A_i$ are 0,1,-1 matrices of size $n \times n$.

It is straightforward to show from the properties of ODs that

1. $A_i A_i^T = s_i I_n$ and

2. $A_i A_j^T + A_j A_i^T = O_n$ for $i \neq j$ where $O_n$ is a matrix of size $n \times n$ with all entries 0.

The program checkOD.py can be used to check ODs in this json format by generating the $A_i$'s and then checking the above conditions.