Non-linear MRD codes from cones over exterior sets
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By using the notion of d-embedding Γ of a (canonical) subgeometry Σ and of exterior set with respect to the h-secant variety Ω_h(𝒜) of a subset 𝒜, 0 ≤ h ≤ n-1, in the finite projective space PG(n-1,q^n), n ≥ 3, in this article we construct a class of non-linear (n,n,q;d)-MRD codes for any 2 ≤ d ≤ n-1. A code 𝒞_σ,T of this class, where 1∈ T ⊂𝔽_q^* and σ is a generator of Gal(𝔽_q^n|𝔽_q), arises from a cone of PG(n-1,q^n) with vertex an (n-d-2)-dimensional subspace over a maximum exterior set ℰ with respect to Ω_d-2(Γ). We prove that the codes introduced in [Cossidente, A., Marino, G., Pavese, F.: Non-linear maximum rank distance codes. Des. Codes Cryptogr. 79, 597–609 (2016); Durante, N., Siciliano, A.: Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries. Electron. J. Comb. (2017); Donati, G., Durante, N.: A generalization of the normal rational curve in PG(d,q^n) and its associated non-linear MRD codes. Des. Codes Cryptogr. 86, 1175–1184 (2018)] are appropriate punctured ones of 𝒞_σ,T and solve completely the inequivalence issue for this class showing that 𝒞_σ,T is neither equivalent nor adjointly equivalent to the non-linear MRD code 𝒞_n,k,σ,I, I ⊆𝔽_q, obtained in [Otal, K., Özbudak, F.: Some new non-additive maximum rank distance codes. Finite Fields and Their Applications 50, 293–303 (2018).].
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