New lower bounds for the Hadwiger numbers of ℓp balls for p < 2

In this note, we derive an asymptotic lower bound for the size of constant weight binary codes that is exponential in the code length, if both the minimum distance and the weight grow in proportion to the code length. We use this bound to find new lower bounds for the Hadwiger and weak Hadwiger numbers of d-dimensional ℓp balls in the case 1 ≤ p < 2.