Linear sets and MRD-codes arising from a class of scattered linearized polynomials

TL;DR

This paper introduces a new class of scattered linearized polynomials over finite fields, leading to novel maximum scattered linear sets and infinite families of MRD-codes with specific parameters.

Contribution

It generalizes a known polynomial construction to arbitrary even dimensions and establishes new scattered linear sets and MRD-code families.

Findings

Constructed new scattered linearized polynomials for all even n ≥ 6.

Produced new maximum scattered linear sets in PG(1, q^n) for n=8,10.

Developed infinite families of MRD-codes with minimum distance n-1.

Abstract

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278 is generalized for any even $n\ge6$ to an $\mathbb F_q$-linear automorphism $\psi(x)$ of $\mathbb F_q^n$ of order $n$. Such $\psi(x)$ and some functional powers of it are proved to be scattered. In particular this provides new maximum scattered linear sets of the projective line $\mathrm{PG}(1,q^n)$ for $n=8,10$. The polynomials described in this paper lead to a new infinite family of MRD-codes in $\mathbb F_q^{n\times n}$ with minimum distance $n-1$ for any odd $q$ if $n\equiv0\pmod4$ and any $q\equiv1\pmod4$ if $n\equiv2\pmod4$.