On Inverses for Quadratic Permutation Polynomials over Integer Rings

TL;DR

This paper establishes a necessary and sufficient condition for the inverse degree of quadratic permutation polynomials over integer rings and provides an explicit algorithm for computing these inverses, enhancing turbo coding system design.

Contribution

It introduces a precise condition for the inverse degree and an algorithm to compute inverse polynomials, advancing the understanding of quadratic permutation polynomials in coding.

Findings

Derived a necessary and sufficient condition for inverse degree

Developed an explicit algorithm for inverse polynomial computation

Improved design methods for turbo coding systems

Abstract

Quadratic permutation polynomial interleavers over integer rings have recently received attention in practical turbo coding systems from deep space applications to mobile communications. In this correspondence, a necessary and sufficient condition that determines the least degree inverse of a quadratic permutation polynomial is proven. Moreover, an algorithm is provided to explicitly compute the inverse polynomials.