# On permutation trinomials of type $x^{2p^s+r}+x^{p^{s}+r} +\lambda x^r$

## TL;DR

This paper characterizes all permutation trinomials of a specific form over finite fields under certain size constraints, extending previous results for the case when p=2 and r=1.

## Contribution

It provides a complete classification of permutation trinomials of the given form over finite fields under the condition $(2p^s+r)^4 < p^t$, extending earlier work for special cases.

## Key findings

- All such permutation trinomials are explicitly determined.
- The classification applies when $(2p^s+r)^4 < p^t$, covering a broad class of finite fields.
- Partial extension of previous results for the case $p=2$, $r=1$.

## Abstract

We determine all permutation trinomials of type $x^{2p^s+r}+x^{p^s+r}+\lambda x^r$ over the finite field $\mathbb{F}_{p^t}$ when $(2p^s+r)^4<p^t$. This partially extends a previous result by Bhattacharya and Sarkar in the case $p=2$, $r=1$.