Balanced independent sets in hypercubes

The n-dimensional hypercube Qn has many maximal independent sets of vertices. We study the cardinality of those maximal independent sets which are balanced , i.e. exactly half of whose vertices have even weight, obtaining both upper and lower bounds for the maximum value. For n ≤ 7 we obtain the exact value. For all odd n, we conjecture that the exact value is 2n−1 − ( n−1 (n−1)/2 ) . We also consider balanced independent subsets of the two middle levels of Qn when n is odd and prove that a simply constructed balanced maximal independent set for that subgraph has the maximum cardinality of any balanced independent set. Its cardinality is 2 · ( n−1 (n−3)/2 ) . Finally, one way in which balanced maximal independent sets in Qn arise is from binary linear codes, and we use Hamming codes to find balanced maximal independent sets of small cardinality.