2-(v, 5; m) spontaneous emission error designs

Quantum jump codes are quantum codes which correct errors caused by quantum jumps. A t-spontaneous emission error design (t-SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. Denote by \(\overline{M}(t,k,v)\) the largest possible dimension m for which a nondegenerate t-(v, k; m) SEED exists. \(\overline{M}(2,3,v)\) has been determined completely, which is based on a great deal of research on large sets of Steiner triple systems and large sets of pairwise disjoint compatible 2-(v, 3, 1) packings. For \(k=4\), the upper bounds on dimensions of 2-(v, 4; m) SEEDs were also demonstrated and the corresponding leave graphs were investigated in Zhou and Chang (J Combin Des 24:439–460, 2016). In this paper we turn our attention to the case \(t=2\) and \(k=5\). We present general upper bounds on the dimensions of 2-(v, 5; m) SEEDs and describe the concrete leave graphs of the 2-SEEDs attaining the stated upper bounds.