Cycle structure of permutation functions over finite fields and their applications

TL;DR

This paper introduces new permutation-based interleavers over finite fields using Möbius and Rédéi functions, analyzing their cycle structures and inverses for applications in coding theory.

Contribution

It presents novel deterministic interleavers based on Rédéi and Möbius functions, including explicit inverse formulas and cycle structure analysis.

Findings

Derived exact inverse formulas for Rédéi functions.

Constructed new interleavers with known cycle structures.

Analyzed cycle structures of Rédéi functions in detail.

Abstract

In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb{F}_q$. For the first time M\"{o}bius and R\'edei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of R\'edei functions an exact formula for the inverse function is derived. The cycle structure of R\'edei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers.