Density of normal binary covering codes

A binary code with covering radius R is a subset C of the hypercube Qn = {0, 1}n such that every x ∈ Qn is within Hamming distance R of some codeword c ∈ C, where R is as small as possible. For a fixed coordinate i ∈ [n], define C b , for b ∈ {0, 1}, to be the set of codewords with a b in the ith position. Then C is normal if there exists an i ∈ [n] such that for any v ∈ Qn, the sum of the Hamming distances from v to C 0 and C (i) 1 is at most 2R+1. We newly define what it means for an asymmetric covering code to be normal, and consider the worst case asymptotic densities ν∗(R) and ν∗ +(R) of constant radius R symmetric and asymmetric normal covering codes, respectively. Using a probabilistic deletion method, and analysis adapted from previous work by Krivelevich, Sudakov, and Vu, we show that both are bounded above by e(R logR + logR + log logR + 4), giving evidence that minimum size constant radius covering codes could still be normal.