How to Have a Wrong Bet in Football Pools

Consider the set Qn = {0, 1, 2}n equipped with the usual Hamming distance. Denoted by T (n) the minimal number of spheres of radius n that covers Qn. The exact values of T (n) are known for n ≤ 8 and in particular T (8) = 44. It is known that for any n ≤ 7 (see [1]) up to equivalence there exists unique covering of Qn. In this paper we show that up to equivalence there exist two coverings of Q8 with 44 elements. Also we improve the best known lower bound T (9) ≥ 66 by showing that T (9) ≥ 67. Since there exists a covering of Q9 with 68 spheres we have 67 ≤ T (9) ≤ 68. The inequality T (9) ≥ 67 implies T (10) ≥ 101, T (11) ≥ 152, T (12) ≥ 228 and T (13) ≥ 342, thus improving the best known lower bounds for 10 ≤ n ≤ 13.