A generalization of the normal rational curve in \(\mathrm {PG}(d,q^n)\) and its associated non-linear MRD codes
A rank distance code is a subset of \(M_{m,n}(F_q)\) where the distance between two matrices is the rank of their difference. The size of an \((m,n,q;s)\)-rank distance code \(\chi\) satisfies the following Singleton type bound:\(|\chi | \leq q^{n(m-s)}\). Codes meeting this bound are known as maximum rank distance codes (MRD codes). Take two distinct points \(A,B\) of a \(d\)-dimensional projective space and a collineation \(\Phi\) between the stars of lines with these two points as vertices such that the map does not send the line containing the points to itself. The authors prove that if \(d=2\) or \(d \geq 3\) and the lines \(\Phi^{-1}(AB), AB, \Phi(AB)\) are not in a common plane, then the set \(C\) of points of intersection of corresponding lines under \(\Phi\) is the union of \(q-1\) scattered \(F_q\) linear sets of rank \(n\) together with \(\{A,B\}\). This set \(C\) is used to construct infinite families of non-linear \((d+1,n,q;d-1)\)-MRD codes, with \(d\leq n-1\).
scientific article; zbMATH DE number 6872536
scientific article