The Optimal Rate of Second-Order Generalized-Covering Codes

The generalized-covering radius was recently proposed as a fundamental property of linear codes. We consider a natural extension of this property to general (not necessarily linear) codes, and provide an asymptotic solution to our problem by finding the optimal rate function of second-order covering codes given a fixed normalized covering radius. We also prove that the fraction of second-order covering codes among codes of sufficiently large rate tends to 1 as the code length tends to ∞.