Fourier Codes and Hartley Codes

TL;DR

This paper introduces real-valued block codes derived from DFT and DHT eigensequences, analyzing their algebraic structure, parameters, and potential for efficient transform computations.

Contribution

It presents a novel class of codes based on eigensequences of DFT and DHT, including generator and parity check matrices for lengths up to 24, and explores their lattice properties.

Findings

Codes include Hamming-Hartley and Golay-Hartley examples.

Parameters like dimension, minimal norm, and density are computed.

Potential application in efficient DHT/DFT computations.

Abstract

Real-valued block codes are introduced, which are derived from Discrete Fourier Transforms (DFT) and Discrete Hartley Transforms (DHT). These algebraic structures are built from the eigensequences of the transforms. Generator and parity check matrices were computed for codes up to block length N=24. They can be viewed as lattices codes so the main parameters (dimension, minimal norm, area of the Voronoi region, density, and centre density) are computed. Particularly, Hamming-Hartley and Golay-Hartley block codes are presented. These codes may possibly help an efficient computation of a DHT/DFT.