On the equivalence of linear sets

• Published in: Designs, codes, and cryptography Vol. 81; no. 2; pp. 269 - 281
• Main Authors: Csajbók, Bence, Zanella, Corrado
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2016
• Subjects: Circuits; Coding and Information Theory; Computer Science; Cryptology; Data Structures and Information Theory; Discrete Mathematics in Computer Science; Information and Communication; Collineation; Finite field; Finite projective space; Subgeometry; Linear set; 51E20
• ISSN: 0925-1022; 1573-7586
• DOI: 10.1007/s10623-015-0141-z

Let L be a linear set of pseudoregulus type in a line ℓ in Σ ∗ = PG ( t - 1 , q t ) , t = 5 or t > 6 . We provide examples of q -order canonical subgeometries Σ 1 , Σ 2 ⊂ Σ ∗ such that there is a ( t - 3 ) -subspace Γ ⊂ Σ ∗ \ ( Σ 1 ∪ Σ 2 ∪ ℓ ) with the property that for i = 1 , 2 , L is the projection of Σ i from center Γ and there exists no collineation ϕ of Σ ∗ such that Γ ϕ = Γ and Σ 1 ϕ = Σ 2 . Condition (ii) given in Theorem  3 in Lavrauw and Van de Voorde (Des Codes Cryptogr 56:89–104, 2010 ) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.