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Generalizing two families of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

Alessandro Giannoni, Giovanni Giuseppe Grimaldi, Giovanni Longobardi, Marco Timpanella

TL;DR

The paper advances the theory of scattered $q^s$-linearized polynomials by introducing the quadrinomials $\psi_{m,h,s}$ in the form $\psi_{m,h,s}= m(X^{q^s}-h^{1-q^{s(t+1)}}X^{q^{s(t+1)}})+X^{q^{s(t-1)}}+h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}$ with $n=2t$, $\gcd(s,2t)=1$, and $(m,h)\in \bF_{q^t}\times \bF_{q^{2t}}$, proving they are scattered under broad, explicit conditions that generalize prior families. The approach relies on a detailed decomposition of $\bF_{q^{2t}}$ into $\ker L_m\oplus\ker M$, a split of the scattering condition into a pair of coupled linear systems, and a careful case analysis based on $\mathrm{N}_{q^{2t}/q^t}(h)=-1$ or $+1$, yielding new infinite families beyond the pseudoregulus and LP-type polynomials. The authors also address equivalence, showing the new family is not GL-equivalent to the known families, and compute the stabilizer $G^{\circ}_{\psi}$ of the associated scattered subspace, with the right idealizer isomorphic to $\bF_{q^2}$; they provide both sufficient conditions for scatteredness and partial necessary conditions, plus an appendix on adjoint-equivalence. Collectively, the results expand the catalog of maximum scattered linear sets and MRD codes, while clarifying the landscape of equivalences among known families and the new quadrinomials.

Abstract

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $\mathbb{F}_{q^n}[X]$ are known to exist for infinitely many values of $n$ and $q$: (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $ψ_{m,h,s}$, are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.

Generalizing two families of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

TL;DR

The paper advances the theory of scattered -linearized polynomials by introducing the quadrinomials in the form with , , and , proving they are scattered under broad, explicit conditions that generalize prior families. The approach relies on a detailed decomposition of into , a split of the scattering condition into a pair of coupled linear systems, and a careful case analysis based on or , yielding new infinite families beyond the pseudoregulus and LP-type polynomials. The authors also address equivalence, showing the new family is not GL-equivalent to the known families, and compute the stabilizer of the associated scattered subspace, with the right idealizer isomorphic to ; they provide both sufficient conditions for scatteredness and partial necessary conditions, plus an appendix on adjoint-equivalence. Collectively, the results expand the catalog of maximum scattered linear sets and MRD codes, while clarifying the landscape of equivalences among known families and the new quadrinomials.

Abstract

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in are known to exist for infinitely many values of and : (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by , are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.
Paper Structure (13 sections, 18 theorems, 209 equations)

This paper contains 13 sections, 18 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Proof of Theorem \ref{['main']}
  4. Case N(h) = -1
  5. Case N(h) = 1
  6. Equivalence issue
  7. The stabilizer of U_psi
  8. Necessary conditions
  9. Case (a):
  10. Case (b): $\ell + s \equiv 0 \pmod{2t}$.
  11. Case (c): $\ell \equiv s \pmod{2t}$.
  12. Case (d): $\ell \equiv t-s \pmod{2t}$.
  13. Case (e): $\ell \equiv t+s \pmod{2t}$.

Key Result

Theorem 1.1

Let $t \geq 3$, $q$ an odd prime power, $1 \leq s \leq 2t-1$ an integer such that $\gcd(s,2t)=1$, and $(m,h) \in \mathbb F_{q^t} \times \mathbb F_{q^{2t}}$. The polynomial $\psi_{m,h,s}$ is scattered under one of the following assumptions:

Theorems & Definitions (34)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Proposition 2.7
  • ...and 24 more