Delete some bits and count
Python 2.7 / Gurobi n=9

This solution is very straight usage of Gurobi's ILP solver for the boolean Mixed-Integer Problems (MIP).

The only trick is to take out symmetry in 1's complement to halve the problem sizes.

Using Gurobi LLC's limited time "free" licence we are restricted to 2000 constraints, but solving 10 del 1 is well outside the 60 second time-limit anyway on my laptop.

from gurobipy import *
from itertools import combinations

def mincover(n,d):
    bs = pow(2,n-1-d)
    m = Model()
    m.Params.outputFlag = 0
    b = {}
    for i in range(bs):
      b[i] = m.addVar(vtype=GRB.BINARY, name="b%d" % i)
    m.update()
    for row in range(pow(2,n-1)):
      x = {}
      for i in combinations(range(n), n-d):
        v = 0
        for j in range(n-d):
          if row &amp; pow(2,i[j]):
            v += pow(2,j)
        if v &gt;= bs:
          v = 2*bs-1-v
        x[v] = 1
      m.addConstr(quicksum(b[i] for i in x.keys()) &gt;= 1)
    m.setObjective(quicksum(b[i] for i in range(bs) ), GRB.MINIMIZE)
    m.optimize()
    return int(round(2*m.objVal,0))

for n in range(4,10):
    for d in range((n//2)+1):
        print n, d, mincover(n,d)


UPDATE+CORR: 10,2 has optimal solution size 31 (see e.g.) Gurobi shows no symmetric solution of size 30 exists (returns problem infeasible) .. [my attempt to show asymmetric feasibility at 30 remained inconclusive after 9.5hrs runtime]
e.g. bit patterns of integers 0 7 13 14    25 28 35 36 49 56 63 64 95 106 118 128 147 159 170 182 195 196 200 207 225 231 240 243 249 252 255 or 0 7 13 14 19 25 28 35 36 49 56 63 64 95 106 118 128     159 170 182 195 196 200 207 225 231 240 243 249 252 255