A Large Family of Maximum Scattered Linear Sets of PG(1,qn) and Their Associated MRD Codes

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 PG ( 1 , q n ) . In this paper, we provide a large family of new maximum scattered linear sets over PG ( 1 , q n ) for any even n ≥ 6 and odd q . In particular, the relevant family contains at least q t + 1 8 r t , if t ≢ 2 ( mod 4 ) ; q t + 1 4 r t ( q 2 + 1 ) , if t ≡ 2 ( mod 4 ) , inequivalent members for given q = p r and n = 2 t > 8 , where p = char ( 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 PG ( 1 , q n ) in all classes known so far, is smaller than q 2 ϕ ( n ) / 2 , where ϕ 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.