2024/08/22 by Valentino Smaldore, Corrado Zanella, Ferdinando Zullo
Computer Science · Mathematics · Engineering · #Coding theory and cryptography #Finite Group Theory Research #graph theory and CDMA systems
paper · doi:10.1016/j.laa.2024.08.012
Let 1 < t < n be integers, where t is a divisor of n . An R- q t -partially scattered polynomial is a q -polynomial f in F q n [ X ] that satisfies the condition that for all x , y ∈ F q n ⁎ such that x / y ∈ F q t , if f ( x ) / x = f ( y ) / y , then x / y ∈ F q ; f is called scattered if this implication holds for all x , y ∈ F q n ⁎ . Two polynomials in F q n [ X ] are said to be equivalent if their graphs are in the same orbit under the action of the group Γ L ( 2 , q n ) . For n > 8 only three families of scattered polynomials in F q n [ X ] are known: ( i ) monomials of pseudoregulus type, ( i i ) binomials of Lunardon-Polverino type, and ( i i i ) a family of quadrinomials defined in [1] , [10] and extended in [8] , [13] . In this paper we prove that the polynomial φ m , q J = X q J ( t − 1 ) + X q J ( 2 t − 1 ) + m ( X q J − X q J ( t + 1 ) ) ∈ F q 2 t [ X ] , q odd, t ≥ 3 is R- q t -partially scattered for every value of m ∈ F q t ⁎ and J coprime with 2 t . Moreover, for every t > 4 and q > 5 there exist values of m for which φ m , q is scattered and new with respect to the polynomials mentioned in ( i ), ( i i ) and ( i i i ) above. The related linear sets are of Γ L -class at least two.