A new family of maximum scattered linear sets in PG(1, q^6)

Abstract

The exploration of linear subspaces, particularly scattered subspaces, has garnered considerable attention across diverse mathematical disciplines in recent years, notably within finite geometries and coding theory. Scattered subspaces play a pivotal role in analyzing various geometric structures such as blocking sets, two-intersection sets, complete arcs, caps in affine and projective spaces over finite fields and rank metric codes. This paper introduces a new infinite family of h-subspaces, along with their associated MRD codes. Additionally, it addresses the task of determining the generalized weights of these codes. Notably, we demonstrate that these MRD codes exhibit some larger generalized weights compared to those previously identified.

Exceptional scattered polynomials

“…with q odd, gcd(s, 2t) = gcd(k, 2t) = 1 and 0 ≤ s, k ≤ 2t − 1, see [24,Proposition 2.18]. This family generalizes the example in [25], which for 2t = 6 may be rewritten as in (9), see [7,Proposition 3.9] and see also [40].…”

Extending two families of maximum rank distance codes

“…Notice that this assumption on t is taken in order to ease the computations, even though the codes C h,t,σ are MRD also for t = 3, 4. When t = 3 the inequivalence with the other known F q n -linear MRD codes such as Gabidulin codes, twisted Gabidulin codes and those in [6,7,10,28,47] has been proved in [3,Section 4]. However when t ∈ {3, 4} the computations of this section become more complicated, since some of the arguments that we are going to use do not work.…”

“…Further examples of F q n -linear MRD codes can be found in [3,6,7,10,28,47] which exist only for n ∈ {6, 7, 8} and in [23,24] which exist for every n even.…”

“…Another large family of MRD codes was later given by Trombetti and Zhou in [45]. Except from that, almost all the other known MRD construction have all very specific restriction on some of the parameters: for instance, the MRD codes [3,6,7,10,28,47] exist only for n ∈ {6, 7, 8}.…”

Linear maximum rank distance codes of exceptional type

“…x q si : i / ∈ {0, 1, t − 1, t + 1, 2t − 1}, h1(x) = x q s − x q s(t−1) , h2(x) = δ q t +1 x q s − x q s(t+1) , h3(x) = δ 1−q 2t−1 x q s − x q s(2t−1) q odd, N q 2t (q t (δ) = −1, gcd(s, n) = 1 [7,24,25,31,40] 6 2…”