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On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors

Maximilien Gadouleau, Luca Mariot, Federico Mazzone

TL;DR

The work investigates maximal families of monic degree-$n$ polynomials with nonzero constant term over $\mathbb{F}_q$ under the constraint that pairwise gcd degrees do not exceed $d$, a problem tied to minimum distance in subspace codes derived from linear cellular automata. It first provides a general lower-bound construction for any admissible $d < n/2$, yielding a cardinality lower bound expressed in terms of irreducible counts $I_i$. Specializing to the binary field with $d=1$, the authors present a maximal construction with cardinality $N_n = \sum_{i=1}^{\left\lfloor n/2 \right\rfloor} I_i + I_{n-1} + I_n$ and prove its maximality via an injective mapping argument, then give a full characterization of all maximal families in this case. These results connect polynomial factorization structure to subspace-code design and raise questions for extending the analysis to larger $d$ and $q$, as well as counting the number of maximal families.

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.

On Maximal Families of Binary Polynomials with Pairwise Linear Common Factors

TL;DR

The work investigates maximal families of monic degree- polynomials with nonzero constant term over under the constraint that pairwise gcd degrees do not exceed , a problem tied to minimum distance in subspace codes derived from linear cellular automata. It first provides a general lower-bound construction for any admissible , yielding a cardinality lower bound expressed in terms of irreducible counts . Specializing to the binary field with , the authors present a maximal construction with cardinality and prove its maximality via an injective mapping argument, then give a full characterization of all maximal families in this case. These results connect polynomial factorization structure to subspace-code design and raise questions for extending the analysis to larger and , as well as counting the number of maximal families.

Abstract

We consider the construction of maximal families of polynomials over the finite field , all having the same degree and a nonzero constant term, where the degree of the GCD of any two polynomials is with . 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 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 . There, we characterize the maximal families of polynomials over the binary field . Our findings prompt several more open questions, which we plan to address in an extended version of this work.
Paper Structure (6 sections, 1 theorem, 6 equations)

This paper contains 6 sections, 1 theorem, 6 equations.

Table of Contents

  1. Background and Problem Statement
  2. Lower Bound in the General Case
  3. Maximal Family for Linear Common Factor
  4. Proof:
  5. Characterization of maximal families
  6. Conclusions and Future Works

Key Result

Theorem 3.1

In $\mathbb{F}_2$, a maximal element of $\mathcal{M}_n^1$ has cardinality

Theorems & Definitions (1)

  • Theorem 3.1