Binary Parseval frames from group orbits

TL;DR

This paper investigates binary Parseval frames generated by group orbits, characterizing their structure through algebraic and convolution properties, and applies these findings to optimize error correction codes using abelian groups.

Contribution

It provides a classification of binary Parseval group frames via convolution identities and explores their application in designing effective error correction codes.

Findings

Binary Parseval frames are characterized by symmetric idempotent Gramian matrices.

Equivalence classes of such frames correspond to binary functions satisfying convolution identities.

Using abelian groups like Z_p^q yields better error correction codes than Z_{p^q}.

Abstract

Binary Parseval frames share many structural properties with real and complex ones. On the other hand, there are subtle differences, for example that the Gramian of a binary Parseval frame is characterized as a symmetric idempotent whose range contains at least one odd vector. Here, we study binary Parseval frames obtained from the orbit of a vector under a group representation, in short, binary Parseval group frames. In this case, the Gramian of the frame is in the algebra generated by the right regular representation. We identify equivalence classes of such Parseval frames with binary functions on the group that satisfy a convolution identity. This allows us to find structural constraints for such frames. We use these constraints to catalogue equivalence classes of binary Parseval frames obtained from group representations. As an application, we study the performance of binary Parseval frames generated with abelian groups for purposes of error correction. We show that $Z_p^q$ is always preferable to $Z_{p^q}$ when searching for best performing codes associated with binary Parseval group frames.