What is odd about binary Parseval frames?

TL;DR

This paper investigates the unique properties and construction challenges of binary Parseval frames, focusing on their existence of complementary frames and characterization of Gram matrices, especially circulant ones.

Contribution

It provides new insights and an algorithm for constructing binary orthonormal sequences and characterizes binary Parseval frames with circulant Gram matrices.

Findings

Not all binary Parseval frames have complementary frames.

The paper offers an algorithm for constructing binary orthonormal sequences.

Characterization of circulant Gram matrices of binary Parseval frames.

Abstract

This paper examines the construction and properties of binary Parseval frames. We address two questions: When does a binary Parseval frame have a complementary Parseval frame? Which binary symmetric idempotent matrices are Gram matrices of binary Parseval frames? In contrast to the case of real or complex Parseval frames, the answer to these questions is not always affirmative. The key to our understanding comes from an algorithm that constructs binary orthonormal sequences that span a given subspace, whenever possible. Special regard is given to binary frames whose Gram matrices are circulants.