CRT and Fixed Patterns in Combinatorial Sequences

TL;DR

This paper introduces a CRT-based framework for analyzing combinatorial sequence generators, revealing fixed patterns and cyclic structures in LFSR sequences, and offers a new method for computing DFT spectra in finite fields.

Contribution

It presents a novel CRT-based approach for structural analysis of LFSR sequences and a method for direct DFT spectral computation in higher finite fields.

Findings

CRT reveals fixed patterns in LFSR sequences

New method for DFT spectral computation in finite fields

Analysis demonstrated on combiner generators

Abstract

In this paper, new context of Chinese Remainder Theorem (CRT) based analysis of combinatorial sequence generators has been presented. CRT is exploited to establish fixed patterns in LFSR sequences and underlying cyclic structures of finite fields. New methodology of direct computations of DFT spectral points in higher finite fields from known DFT spectra points of smaller constituent fields is also introduced. Novel approach of CRT based structural analysis of LFSR based combinatorial sequence is given both in time and frequency domain. The proposed approach is demonstrated on some examples of combiner generators and is scalable to general configuration of combiner generators.