A new construction of anticode-optimal Grassmannian codes

Abstract

In this paper, we consider the well-known unital embedding from F q k into M k (Fq) seen as a map of vector spaces over Fq and apply this map in a linear block code of rate ρ/ over F q k . This natural extension gives rise to a rank-metric code with k rows, k columns, dimension ρ and minimum distance k that satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length n − k. The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code with length n, cardinality q n −1 q k −1 , minimum injection distance k and dimension k that satisfies the anticode upper bound can be constructed.