New Bounds for the Optimal Density of Covering Single-Insertion Codes Using Turán Density

The density of any covering single-insertion code over an n-symbol alphabet cannot be smaller than 1/r + δr for some positive real δr not depending on n, improving the previous lower bound of 1/(r+1). The asymptotic upper bound can also be improved from 7/(r+1) to 4.911/(r+1) for sufficiently large r.