$r$-fat linearized polynomials over finite fields

Abstract

In this paper we prove that the property of being scattered for a F q -linearized polynomial of small q-degree over a finite field F q n is unstable, in the sense that, whenever the corresponding linear set has at least one point of weight larger than one, the polynomial is far from being scattered. To this aim, we define and investigate r-fat polynomials, a natural generalization of scattered polynomials. An r-fat F q -linearized polynomial defines a linear set of rank n in the projective line of order q n with r points of weight larger than one. When r equals 1, the corresponding linear sets are called clubs, and they are related with a number of remarkable mathematical objects like KM-arcs, group divisible designs and rank metric codes. Using techniques on algebraic curves and global function fields, we obtain numerical bounds for r and the non-existence of exceptional r-fat polynomials with r > 0. In the case n ≤ 4, we completely determine the spectrum of values of r for which an r-fat polynomial exists. In the case n = 5, we provide a new family of 1-fat polynomials. Furthermore, we determine the values of r for which the so-called LP-polynomials are r-fat.

Linear sets on the projective line with complementary weights

“…We conclude the paper by showing in Appendix A the list of the q-polynomials defining F q -linear sets in PG(1, q 4 ), which was an open problem stated in [16] and in [6].…”

“…The authors also showed that F q -linear blocking sets in PG(2, q n ) are simple, that is if L U and L W are two PΓL(3, q n )-equivalent F q -linear blocking sets in PG(2, q n ) if and only U and W are ΓL(3, q n )equivalent. So that, from the above mentioned classification follows the classification of F q -linear set of rank 4 in PG(1, q 4 ); see also [6,Corollary 5.4].…”

“…In [15,5. Section 6] and in [6,Open Problem 8.4] was stated the problem of determining the polynomial shape of the F q -linear sets of rank 4 of PG (1, q 4 ). This section is devoted to explicitly solve this problem.…”

Optimal Selection for Good Polynomials of Degree up to Five

“…Then the decomposition group D(R|P ) is either C 4 or C 2 . In the former case, for some t 0 ∈ F q the polynomial f − t 0 factors as (X − a) 4 , and this leads to a contradiction as in the proof of Lemma 3.2. In the latter case, up to translations we have f − t 0 = X 2 (X − a) 2 for some a, t 0 ∈ F q with a = 0, because the element of order 2 in C 4 is a product of two disjoint transpositions.…”

“…The machinery we use involves Galois theory, the classification of transitive subgroups of the symmetric group S n up to n = 5, and the theory of function fields, using the results and techniques of [15,16], then further developed in [1,2,4,5,8,9].…”