46 research outputs found
Equivalence and automorphism groups of two families of maximum scattered linear sets
Linear set in projective spaces over finite fields plays central roles in the
study of blocking sets, semifields, rank-metric codes and etc. A linear set
with the largest possible cardinality and the maximum rank is called maximum
scattered. Despite two decades of study, there are only a few number of known
maximum scattered linear sets in projective lines, including the family
constructed by Csajb\'ok, Marino, Polverino and Zanella 2018, and the family
constructed by Csajb\'ok, Marino, Zullo 2018 (also Marino, Montanucci, and
Zullo 2020). This paper aims to solve the equivalence problem of the linear
sets in each of these families and to determine their automorphism groups.Comment: This is an English translation of the original Chinese version
published in Scientia Sinica Mathematic
A new family of MRD-codes
We introduce a family of linear sets of PG(1,q^2n) arising from maximum scattered linear sets of pseudoregulus type of PG(3,q^n). For n=3,4 and for certain values of the parameters we show that these linear sets of PG(1,q^2n) are maximum scattered and they yield new MRD-codes with parameters (6,6,q;5) for q>2 and with parameters (8,8,q;7) for q odd
Field reduction and linear sets in finite geometry
Based on the simple and well understood concept of subfields in a finite
field, the technique called `field reduction' has proved to be a very useful
and powerful tool in finite geometry. In this paper we elaborate on this
technique. Field reduction for projective and polar spaces is formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields