A geometric proof of the gap theorem (Q1268611)

The author studies partitions of the Hamming cube \(\{ 0, 1\}^d\) into a small number of balls (in the Hamming metric). There are some trivial possibilities for such partitions: The whole space is one ball of radius \(d\); if \(p_1, p_2\) are two antipodal points and \(r_1, r_2\) radii with \(r_1 + r_2 = d-1\), then the two balls around \(p_i\) with radius \(r_i\) form a partition of the space. Also taking one ball of radius \(d-2\) and covering the remaining \(d+1\) points by balls of radius \(0\) gives a partition of the space in \(d+2\) balls. The author proves that these are the only small such partitions: if \(p\) is the largest prime \(p\leq d\), then any partition of \(\{ 0,1\}^d\) into balls which is not of one of these three types requires at least \(2p+2\) balls. This extends a previous result of \textit{H. D. L. Hollmann, J. Körner} and \textit{S. Litsyn} [J. Comb. Theory, Ser. A 80, No. 2, 388-393, Art. No. TA972816 (1997; Zbl 0883.05036)].