The rank of a scattered Fq-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 Fq-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 Fq-linear sets of PG (1 , q^n) of maximum rank n yield Fq-linear MRD-codes with dimension 2n and minimum distance n- 1. We generalize this result and show that scattered Fq-linear sets of PG (r- 1 , q^n) of maximum rank rn / 2 yield Fq-linear MRD-codes with dimension rn and minimum distance n- 1.
Maximum scattered linear sets and MRD-codes / Csajbok, Bence; Marino, Giuseppe; Polverino, Olga; Zullo, Ferdinando. - In: JOURNAL OF ALGEBRAIC COMBINATORICS. - ISSN 0925-9899. - STAMPA. - 46:3-4(2017), pp. 517-531. [10.1007/s10801-017-0762-6]
Maximum scattered linear sets and MRD-codes
Csajbok, Bence;
2017-01-01
Abstract
The rank of a scattered Fq-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 Fq-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 Fq-linear sets of PG (1 , q^n) of maximum rank n yield Fq-linear MRD-codes with dimension 2n and minimum distance n- 1. We generalize this result and show that scattered Fq-linear sets of PG (r- 1 , q^n) of maximum rank rn / 2 yield Fq-linear MRD-codes with dimension rn and minimum distance n- 1.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
