A Large Family of Maximum Scattered Linear Sets of ${{\,\mathrm{{PG}}\,}}(1,q^n)$ and Their Associated MRD Codes

Abstract

Linear sets in projective spaces over finite fields were introduced by Lunardon (Geom Dedic 75(3):245–261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric codes, 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 $n\ge 6$ and odd q. In particular, the relevant family contains at least

$$\begin{aligned} {\left\{ \begin{array}{ll} \left\lfloor \frac{q^t+1}{8rt}\right\rfloor ,&{} \text { if }t\not \equiv 2\pmod {4};\\ \left\lfloor \frac{q^t+1}{4rt(q^2+1)}\right\rfloor ,&{} \text { if }t\equiv 2\pmod {4}, \end{array}\right. } \end{aligned}$$

inequivalent members for given $q=p^r$ and $n=2t>8$, where $p=\textrm{char}({{\mathbb {F}}}_q)$. 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 $q^2\phi (n)/2$, where $\phi$ denotes Euler’s totient function. Moreover, we show that there are a large number of new maximum rank-distance codes arising from the constructed linear sets.