Circulant Matrix Representation of PN-sequences with Ideal Autocorrelation Property

TL;DR

This paper explores the properties of PN-sequences with ideal autocorrelation, introduces a circulant matrix representation, and derives conditions for their structure using algebraic and combinatorial methods.

Contribution

It presents a novel circulant matrix framework for analyzing PN-sequences with ideal autocorrelation and derives new algebraic conditions for their structure.

Findings

Circulant matrix representation simplifies analysis of PN-sequences.

Derived algebraic conditions for sequences with ideal autocorrelation.

Connected autocorrelation properties to Hamming weight and distance.

Abstract

In this paper, we investigate PN-sequences with ideal autocorrelation property and the consequences of this property on the number of +1s and -1s and run structure of sequences. We begin by discussing and surveying about the length of PNsequences with ideal autocorrelation property. From our discussion and survey we introduce circulant matrix representation of PN-sequence. Through circulant matrix representation we obtain system of non-linear equations that lead to ideal autocorrelation property. Rewriting PN-sequence and its autocorrelation property in {0,1} leads to a definition based on Hamming weight and Hamming distance and hence we can easily prove some results on the PN-sequences with ideal autocorrelation property.