Maximum scattered linear sets and MRD-codes

• Published in: Journal of algebraic combinatorics Vol. 46; no. 3-4; pp. 517 - 531
• Main Authors: Csajbók, Bence, Marino, Giuseppe, Polverino, Olga, Zullo, Ferdinando
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2017; Springer Nature B.V
• Subjects: Codes; Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Linear set; Scattered spaces; MRD-codes
• ISSN: 0925-9899; 1572-9192; 1572-9192
• DOI: 10.1007/s10801-017-0762-6

The rank of a scattered F q -linear set of PG ( r - 1 , q n ) , rn even, is at most rn  / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r , n , q ( rn even) for scattered F q -linear sets of rank rn  / 2. In this paper, we prove that the bound rn  / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered F q -linear sets of PG ( 1 , q n ) of maximum rank n yield F q -linear MRD-codes with dimension 2 n and minimum distance n - 1 . We generalize this result and show that scattered F q -linear sets of PG ( r - 1 , q n ) of maximum rank rn  / 2 yield F q -linear MRD-codes with dimension rn and minimum distance n - 1 .