Application of Generating Functions and Partial Differential Equations in Coding Theory

TL;DR

This paper explores the use of generating functions and partial differential equations in coding theory, analyzing solution behaviors and introducing complex analysis techniques to improve error probability analysis.

Contribution

It introduces a novel application of Hadamard multiplication in analyzing power sums related to error probabilities in coding theory.

Findings

Solutions for Cycle Poisson PDEs show rapid growth tendencies.

Simulation of coefficients reveals significant growth patterns.

Hadamard multiplication offers a new analytical approach.

Abstract

In this work we have considered formal power series and partial differential equations, and their relationship with Coding Theory. We have obtained the nature of solutions for the partial differential equations for Cycle Poisson Case. The coefficients for this case have been simulated, and the high tendency of growth is shown. In the light of Complex Analysis, the Hadamard Multiplication's Theorem is presented as a new approach to divide the power sums relating to the error probability, each part of which can be analyzed later.