From a linear block code B over the Galois ring GR(4, m) with a k x n generator matrix and minimum Hamming distance d, a rate-k/n convolutional code over the ring 4 with squared Euclidean free distance at least 2d and a non-recursive encoder with memory at most m -1 is constructed. When the generator matrix of B is systematic, the convolutional encoder is systematic, basic, non-catastrophic and minimal. Long codes constructed in this manner are shown to satisfy a GilbertVarshamov bound.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.