On the Construction of Finite Oscillator Dictionary

TL;DR

This paper revisits the finite split oscillator dictionary with simplified proofs, explores non-split tori in group theory, and provides an explicit algorithm for constructing the finite non-split oscillator dictionary, advancing sequence design and compressive sensing applications.

Contribution

It offers a simplified proof of the finite split oscillator dictionary, studies non-split tori in SL(2, F_p), and introduces an explicit algorithm for the non-split oscillator dictionary.

Findings

Simplified proof of the finite split oscillator dictionary.

Analysis of non-split tori in SL(2, F_p).

Explicit algorithm for the non-split oscillator dictionary.

Abstract

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani and Sochen. In this paper, we first revisit closed formulae of the finite split oscillator dictionary $\mathfrak{S}^s$ by a simple proof. Then we study the non-split tori of the group $SL(2,\mathbb{F}_p)$. Finally, An explicit algorithm for computing the finite non-split oscillator dictionary $\mathfrak{S}^{ns}$ is described.