The maximum scattered linear sets in $PG(1,q^n)$ have been completely classified for $n \le 4$ by Csajbók-Zanella and Lavrauw-Van de Voorde. Here a wide class of linear sets in $PG(1,q^5)$ is studied which depends on two parameters. Conditions for the existence, in this class, of possible new maximum scattered linear sets in $PG(1,q^5)$ are exhibited.
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