Hadamard matrices and Legendre pairs

Hadamard Matrices (HMs) have been studied for more than 130 years.

The main existence conjecture for HMs, remains open, 

i.e. that there exists an HM of order n, for every n which is a multiple of 4.  

Legendre Pairs (LPs) are interesting combinatorial objects 

that can be used to construct Hadamard matrices. 

The main existence conjecture for LPs, remains open, 

i.e. that there exists a LP of order l, for every odd l. 

The existence of a LP of order l, implies the existence of a Hadamard 

matrix of order 2l+2, therefore, the existence conjecture for LPs, implies 

the existence conjecture for HMs, albeit via a more "structured" form.

We will summarize old and new approaches and results in the study of 

Legendre Pairs, with an emphasis on computational aspects and open problems.