On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors

TL;DR

This paper investigates the construction and characterization of maximal families of binary polynomials with pairwise linear common factors, linking their properties to subspace codes and minimum distance in finite fields.

Contribution

It provides a lower bound on the size of such polynomial families and characterizes the maximal families over  for the case where the GCD degree is one.

Findings

Established a lower bound on the cardinality of polynomial families

Characterized maximal families over  for GCD degree one

Connected polynomial families to subspace code minimum distance

Abstract

We consider the construction of maximal families of polynomials over the finite field $\mathbb{F}_q$, all having the same degree $n$ and a nonzero constant term, where the degree of the GCD of any two polynomials is $d$ with $1 \le d\le n$. The motivation for this problem lies in a recent construction for subspace codes based on cellular automata. More precisely, the minimum distance of such subspace codes relates to the maximum degree $d$ of the pairwise GCD in this family of polynomials. Hence, characterizing the maximal families of such polynomials is equivalent to determining the maximum cardinality of the corresponding subspace codes for a given minimum distance. We first show a lower bound on the cardinality of such families, and then focus on the specific case where $d=1$. There, we characterize the maximal families of polynomials over the binary field $\mathbb{F}_2$. Our findings prompt several more open questions, which we plan to address in an extended version of this work.