Linearized trinomials with maximum kernel

TL;DR

This paper derives explicit formulas for linearized trinomials over finite fields with maximum kernel dimension, providing new constructions and characterizations relevant to coding theory and algebraic geometry.

Contribution

It offers closed-form formulas for coefficients of $\sigma$-trinomials with maximum kernel, characterizes those with degree 3 and 4, and applies results to rank metric codes and subspace codes.

Findings

Explicit formulas for $\sigma$-trinomials with maximum kernel.

Characterization of degree 3 and 4 cases.

New constructions for rank metric and subspace codes.

Abstract

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $\sigma$ be a generator of $\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. In this paper we provide closed formulas for the coefficients of a $\sigma$-trinomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the dimension of the kernel of $f$ equals its $\sigma$-degree, that is linearized polynomials with maximum kernel. As a consequence, we present explicit examples of linearized trinomials with maximum kernel and characterize those having $\sigma$-degree $3$ and $4$. Our techniques rely on the tools developed in [24]. Finally, we apply these results to investigate a class of rank metric codes introduced in [8], to construct quasi-subfield polynomials and cyclic subspace codes, obtaining new explicit constructions to the conjecture posed in [37].