Maximum scattered linear sets and MRD-codes

Abstract

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 2n 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.

13. MRD codes: constructions and connections

“…In [93], a connection between certain MRD codes and scattered linear sets on a projective line was made. This was extended in different ways in [17] and [94]. Theorem 9 ([93]).…”

“…Theorem 9 ([93]). Equivalence classes of maximum scattered subspaces with respect to the Desarguesian n-spread in V (2n, q) are in one-to-one correspondence with equivalence classes of F q n -linear MRD codes in M n×n (F q ) with minimum distance n − 1 Theorem 10 ( [17]). F q -linear MRD codes in M sn 2 ,n (F q ) with minimum distance n − 1 can be constructed from scattered subspaces of F q -dimension sn 2 in V (s, q n ).…”

On the sparseness of certain linear MRD codes

“…The latter codes are even F q m -linear when identifying the ambient matrix space F m×n q with F n q m . Further constructions of MRD codes (not F q m -linear in general) have been found and studied by Sheekey [19], Cossidente et al [3], Csajbók et al [4], Lunardon et al [15], Trombetti/Zhou [23] and others.…”

Puncturing maximum rank distance codes

“…, m − 1}, the punctured code P A (H (k) n,n,t,µ,s ) is an MRD code with the same parameters as the code Φ (k) m,n,t defined by (10). We denote such (m, n, q; m − t + 1)-MRD code by H m,n,t,µ,s has the same parameters as the (m, rm/2, q; m − 1)-MRD codes provided in [9] only for t = 2 and n = m.…”

“…In this paper the term "generalized twisted Gabidulin code" will be used for codes defined in [30, Remark 8]. For different relations between linear MRD codes and linear sets see [9,22], [30, Section 5], [7, Section 5]. To the extent of our knowledge, these are the only infinite families of linear MRD codes with m < n appearing in the literature.In [12] infinite families of non-linear (n, n, q; n − 1)-MRD codes, for q ≥ 3 and n ≥ 3 have been constructed.…”