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 M ¯ ( t , k , v ) the largest possible dimension m for which a nondegenerate t -( v , k ; m ) SEED exists. 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.