A large family of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ and their associated MRD codes

TL;DR

This paper constructs a large family of maximum scattered linear sets in projective lines over finite fields, significantly increasing known examples and deriving new maximum rank-distance codes, with implications for finite geometry and coding theory.

Contribution

It introduces a broad new family of maximum scattered linear sets in PG(1,q^n) for even n≥6 and odd q, vastly expanding the known examples and their associated MRD codes.

Findings

Constructed a large family of maximum scattered linear sets for specified parameters.

Provided lower bounds on the number of inequivalent linear sets in the family.

Showed the derived linear sets lead to many new MRD codes.

Abstract

The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line $\mathrm{PG}(1,q^n)$. In this paper, we provide a large family of new maximum scattered linear sets over $\mathrm{PG}(1,q^n)$ for any even $n\geq 6$ and odd $q$. In particular, the relevant family contains at least \[ \begin{cases} \left\lfloor\frac{q^t+1}{8rt}\right\rfloor,& \text{ if }t\not\equiv 2\pmod{4};\\[8pt] \left\lfloor\frac{q^t+1}{4rt(q^2+1)}\right\rfloor,& \text{ if }t\equiv 2\pmod{4}, \end{cases} \] inequivalent members for given $q=p^r$ and $n=2t>8$, where $p=\mathrm{char}(\mathbb{F}_q)$. This is a great improvement of previous results: for given $q$ and $n>8$, the number of inequivalent maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ in all classes known so far, is smaller than $q^2$. Moreover, we show that there are a large number of new maximum rank-distance codes arising from the constructed linear sets.