Vectorial FCSR constructed on totally ramified extension of the p-adic numbers

TL;DR

This paper introduces a vectorial approach to d-FCSRs over finite fields, analyzing their structure, properties, and maximal sequences, with comparisons to existing models.

Contribution

It develops a new vectorial framework for d-FCSRs over finite fields, extending previous models and analyzing their fundamental properties.

Findings

Established basic properties like periodicity and maximal length sequences

Presented illustrative examples of vectorial d-FCSRs

Compared new models with existing FCSR constructions

Abstract

In this paper, we introduce a vectorial conception of d-FCSRs to build these registers over any finite field. We describe the structure of d-vectorial FCSRs and we develop an analysis to obtain basic properties like periodicity and the existence of maximal length sequences. To illustrate these vectorial d-FCSRs, we present simple examples and we compare with those of Goresky, Klapper and Xu. Keywords: LFSR, FCSR, vectorial FCSR, d-FCSR, sequences, periodicity, p-adic, ?-adic, maximal period.