Frame theory for binary vector spaces

TL;DR

This paper develops a comprehensive theory of frames and Parseval frames for finite-dimensional binary vector spaces, highlighting differences from classical Hilbert space theory due to the absence of an inner product.

Contribution

It introduces characterizations of binary frames, defines switching equivalence, and classifies all binary Parseval frames in low dimensions.

Findings

Characterizations similar to classical frames are established.

Switching equivalence for binary frames is defined.

All low-dimensional binary Parseval frames are classified.

Abstract

We develop the theory of frames and Parseval frames for finite-dimensional vector spaces over the binary numbers. This includes characterizations which are similar to frames and Parseval frames for real or complex Hilbert spaces, and the discussion of conceptual differences caused by the lack of a proper inner product on binary vector spaces. We also define switching equivalence for binary frames, and list all equivalence classes of binary Parseval frames in lowest dimensions, excluding cases of trivial redundancy.