1,720,993 research outputs found
Saturating linear sets in PG(2,q4)
No full textBonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1-saturating linear sets in PG(2, q4), that is Fq-linear sets in PG(2, q4) with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/)
On subspace designs
Full text linkGuruswami and Xing introduced subspace designs in 2013 to give the first
construction of positive rate rank metric codes list-decodable beyond half the
distance. In this paper we provide bounds involving the parameters of a
subspace design, showing they are tight via explicit constructions. We point
out a connection with sum-rank metric codes, dealing with optimal codes and
minimal codes with respect to this metric. Applications to two-intersection
sets with respect to hyperplanes, two-weight codes, cutting blocking sets and
lossless dimension expanders are also provided
MRD-codes arising from the trinomial x<sup>q</sup>+x<sup>q<sup>3</sup> </sup>+cx<sup>q<sup>5</sup> </sup>∈F<sub>q<sup>6</sup> </sub>[x]
No full textIn [10], the existence of Fq-linear MRD-codes of Fq 6×6, with dimension 12, minimum distance 5 and left idealiser isomorphic to Fq6 , defined by a trinomial of Fq6 [x], when q is odd and q≡0,±1(mod5), has been proved. In this paper we show that this family produces Fq-linear MRD-codes of Fq 6×6, with the same properties, also in the remaining q odd cases, but not in the q even case. These MRD-codes are not equivalent to the previously known MRD-codes. We also prove that the corresponding maximum scattered Fq-linear sets of PG(1,q6) are not PΓL(2,q6)-equivalent to any previously known linear set.</p
Maximum flag-rank distance codes
Full text linkIn this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes
New scattered linearized quadrinomials
Full text linkLet 18 only three families of scattered polynomials in F(q)n[X] are known: (i) monomials of pseudoregulus type, (ii) binomials of Lunardon-Polverino type, and (iii) a family of quadrinomials defined in [1], [10] and extended in [8], [13]. In this paper we prove that the polynomial phi(m,q)J=Xq(J(t-1)) +Xq(J(2t-1))+m(Xq(J)-Xq(J(t+1))) is an element of F-q(2t)[X], q odd, t >= 3 is R-q(t)-partially scattered for every value of m is an element of F-qt* and J coprime with 2t. Moreover, for every t>4 and q>5 there exist values of m for which phi(m,q) is scattered and new with respect to the polynomials mentioned in (i), (ii) and (iii) above. The related linear sets are of Gamma L-class at least two
On the stabilizer of the graph of linear functions over finite fields
No full textIn this paper we will study the action of F q n 2 × 2 on the graph of an F q -linear function of F q n to itself. In particular, we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also give some examples where this is not the case. We will also connect such a stabilizer to the right idealizer of the rank-metric code defined by the linear function, and give some structural results in the case where the polynomials are partially scattered
Identifiers for MRD-codes
Full text linkFor any admissible value of the parameters (n) and (k) there exist ([n,k])-Maximum Rank distance ({mathbb F}_q)-linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank distance codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes
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