Invariant sets under linear operator and covering codes over modules

A connection between sumsets and covering codes of a module

“…A closely related concept is described as follows. A subset H of X n is an R-short covering of X n if each v ∈ X n can be written as (7)…”

“…Let a = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) and b = (1, 1, 1, 1, 1, 1, 1, 2, 2, 2). The set H = {a, b} is a minimal 7-short covering of F 10 5 , which yields C(F 5 , 10, 7) = 2 (see [7] for more general results).…”

“…Eq. (9) has been explored to reach new bounds on K |X| (n, R) from bounds on C(X, n, R), according to [7] and some of its references. For our purpose, we need the next result.…”

Intersecting families of extended balls in the Hamming spaces

“…The covering problem induced by extended balls in an arbitrary space F n q is called short covering, motivated by the fact that short covering might provide us a way to store nonlinear codes using less memory than the classical ones. Applications to the classical numbers K q (n, R) (from covering codes) appear in [15] and some of its references. On theoretical viewpoints, results on short coverings have been obtained from distinct tools: graph theory [11], linear algebra [15], ring theory [14,18].…”

“…Applications to the classical numbers K q (n, R) (from covering codes) appear in [15] and some of its references. On theoretical viewpoints, results on short coverings have been obtained from distinct tools: graph theory [11], linear algebra [15], ring theory [14,18].…”