On the Divisibility of Trinomials by Maximum Weight Polynomials over F2

TL;DR

This paper investigates the divisibility of trinomials by maximum weight polynomials over finite fields, extending previous work to construct orthogonal arrays of strength at least 3.

Contribution

It generalizes the divisibility analysis from pentanomials to maximum weight polynomials over _2, enabling new constructions of orthogonal arrays of strength at least 3.

Findings

Established divisibility conditions for trinomials by maximum weight polynomials.

Applied these results to construct orthogonal arrays of strength at least 3.

Extended previous results from pentanomials to more general polynomial classes.

Abstract

Divisibility of trinomials by given polynomials over finite fields has been studied and used to construct orthogonal arrays in recent literature. Dewar et al.\ (Des.\ Codes Cryptogr.\ 45:1-17, 2007) studied the division of trinomials by a given pentanomial over $\F_2$ to obtain the orthogonal arrays of strength at least 3, and finalized their paper with some open questions. One of these questions is concerned with generalizations to the polynomials with more than five terms. In this paper, we consider the divisibility of trinomials by a given maximum weight polynomial over $\F_2$ and apply the result to the construction of the orthogonal arrays of strength at least 3.