Harmonic index designs in binary Hamming schemes (Q1696524)

We recall the ``addition formula'' for a commutative association scheme. From this formula we construct a linear programming for the size of harmonic index designs and define the notion of tight designs using Fisher type inequality. We examine the existence of tight harmonic index \(T\)-designs for some \(T\) in binary Hamming association schemes \(H(d, 2)\). The non-existence of tight harmonic index \(T\)-designs in binary Hamming schemes \(H(d, 2)\) for \(T = \{4\}, \{6\}, \{8, 4\}, \{8, 2\}, \{6, 4\}\) and \(\{6, 2\}\) was showed and the asymptotic (i.e., \(d \rightarrow \infty\)) non-existence of tight designs of harmonic index \(T = \{2e\}\) for \(e \geq 4\) was proved. A necessary and sufficient condition for the existence of tight harmonic index \(\{2\}\)-designs was provided and some tight harmonic index \(\{4, 2\}\)-designs in \(H(6, 2)\) which are any half part of the tight 5-design in \(H(6, 2)\) was found.