Trades in complex Hadamard matrices

TL;DR

This paper investigates the structure of trades in complex Hadamard matrices, establishing minimum size bounds for trades and characterizing rectangular trades, with a conjecture on the minimal size of all trades.

Contribution

It provides a characterization of rectangular trades in complex Hadamard matrices and proves they contain at least n entries, proposing a conjecture for all trades.

Findings

All trades in real Hadamard matrices contain at least n entries.

Rectangular trades in complex Hadamard matrices contain at least n entries.

Conjecture: all trades in complex Hadamard matrices contain at least n entries.

Abstract

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order $n$ all trades contain at least $n$ entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar $c \neq 1$ to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order $n$ and show that they all contain at least $n$ entries. We conjecture that all trades in complex Hadamard matrices contain at least $n$ entries.