New scattered linearized quadrinomials

TL;DR

The work introduces a quadrinomial-based construction $\varphi_{m,\sigma}=X^{\sigma^{t-1}}+X^{\sigma^{2t-1}}+m(X^{\sigma}-X^{\sigma^{t+1}})$ in $\mathcal{L}_{2t,q}$ to produce $R-\,q^t$-partially scattered $q$-polynomials for odd $q$ and $t\ge3$. It proves the $R$-$q^t$-partial scattering for all admissible parameters and identifies parameter regimes ($t>4$, $q>5$) where some $m$ yield scattered polynomials that are new relative to the known families, with corresponding linear sets having $ ext{ΓL}$-class at least two. A detailed stabilizer analysis of the graphs $U_{m,\sigma}$ under $ ext{GL}(2,q^n)$ is conducted, highlighting when non-diagonal stabilizers appear and how this affects equivalence, via the right idealizer of the associated MRD codes. Finally, the work shows nonequivalence to the established families (pseudoregulus, Lunardon–Polverino, and the adjoint-related (iii$-$b) class) and proves the existence of new scattered polynomials in this family for the specified parameter ranges, enriching MRD-code constructions and the geometry of the resulting linear sets.

Abstract

Let $1<t<n$ be integers, where $t$ is a divisor of $n$. An R-$q^t$-partially scattered polynomial is a $\mathbb F_q$-linearized polynomial $f$ in $\mathbb F_{q^n}[X]$ that satisfies the condition that for all $x,y\in\mathbb F_{q^n}^*$ such that $x/y\in\mathbb F_{q^t}$, if $f(x)/x=f(y)/y$, then $x/y\in\mathbb F_q$; $f$ is called scattered if this implication holds for all $x,y\in\mathbb F_{q^n}^*$. Two polynomials in $\mathbb F_{q^n}[X]$ are said to be equivalent if their graphs are in the same orbit under the action of the group $ΓL(2,q^n)$. For $n>8$ only three families of scattered polynomials in $\mathbb F_{q^n}[X]$ are known: $(i)$~monomials of pseudoregulus type, $(ii)$~binomials of Lunardon-Polverino type, and $(iii)$~a family of quadrinomials defined in [1,10] and extended in [8,13]. In this paper we prove that the polynomial $\varphi_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in\mathbb F_{q^{2t}}[X]$, $q$ odd, $t\ge3$ is R-$q^t$-partially scattered for every value of $m\in\mathbb F_{q^t}^*$ and $J$ coprime with $2t$. Moreover, for every $t>4$ and $q>5$ there exist values of $m$ for which $\varphi_{m,q}$ is scattered and new with respect to the polynomials mentioned in $(i)$, $(ii)$ and $(iii)$ above. The related linear sets are of $ΓL$-class at least two.

Theorems & Definitions (30)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof