Abstract
Let A and B be two points of \(\mathop {\mathrm{PG}}(d,q^n)\) and let \(\Phi \) be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if \(d=2\) or \(d\ge 3\) and the lines \(\Phi ^{-1}(AB), AB, \Phi (AB) \) are not in a common plane, then the set \(\mathcal{C}\) of points of intersection of corresponding lines under \(\Phi \) is the union of \(q-1\) scattered \({\mathbb {F}}_{q}\)-linear sets of rank n together with \(\{A,B\}\). As an application we will construct, starting from the set \(\mathcal{C}\), infinite families of non-linear \((d+1, n, q;d-1)\)-MRD codes, \(d\le n-1\), generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).
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Communicated by J. W. P. Hirschfeld.
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Donati, G., Durante, N. A generalization of the normal rational curve in \(\mathop {\mathrm{PG}}(d,q^n)\) and its associated non-linear MRD codes. Des. Codes Cryptogr. 86, 1175–1184 (2018). https://doi.org/10.1007/s10623-017-0388-7
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DOI: https://doi.org/10.1007/s10623-017-0388-7