Non-linear MRD codes from cones over exterior sets

Let Fq be the finite field of q elements, and let Fm×nq be the vector space of m×n matrices with entries over Fq endowed with the rank distance d(A,B)=rk(A−B). A subset of Fm×nq, including at least two elements, is called a rank distance code. The minimum distance of a code C is naturally defined by d(C)=min{d(A,B)∣A,B∈C,A≠B}. If d(C)=d, we say that C is an (m,n,q;d)-rank distance code. An (m,n,q;d)-rank distance code is additive if it is an additive subgroup of Fm×nq. An additive code is Fq-linear if it is a vector subspace of Fm×nq. A code that is not an Fq-subspace is called a non-linear code. The size of an (m,n,q;d)-rank distance code C satisfies the Singleton-like bound: logq(|C|)≤max{m,n}(min{m,n}−d+1). When this bound is achieved, C is called an (m,n,q;d)-maximum rank distance code, or (m,n,q;d)-MRD code. The first class of non-linear MRD codes consists of the (3,3,q;2)-MRD codes constructed by A. Cossidente, G. Marino and F. Pavese [Des. Codes Cryptogr. 79 (2016), no. 3, 597–609; MR3489760]. These arise from a geometrical context and have been generalized in two steps: first, in [N. Durante and A. Siciliano, Electron. J. Combin. 24 (2017), no. 2, Paper No. 2.33; MR3665566], obtaining a family of (n,n,q;n−1)-MRD codes, n≥3, and then in [G. Donati and N. Durante, Des. Codes Cryptogr. 86 (2018), no. 6, 1175–1184; MR3788898], obtaining a family of (d+1,n,q;d)-MRD codes, 2≤d≤n−1. In the paper under review, the authors further extend the construction by considering a projective cone whose vertex is a proper subspace and whose base is an MRD code in the families listed above. As a result, they obtain a class of non-linear (n,n,q;d)-MRD codes, for any 2≤d≤n−1.