Bounds on the Dimensions of 2‐Spontaneous Emission Error Designs
Junling Zhou, Yanxun Chang · Journal of Combinatorial Designs · 2016
Abstract Quantum jump codes are quantum error‐correcting codes which correct errors caused by quantum jumps. A t‐spontaneous emission error design (t‐SEED) was introduced by Beth et al. in 2003 [T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, and M. Mussinger, A new class of designs which protect against quantum jumps, Des Codes Cryptogr 29 (2003), 51–70.] 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. A nondegenerate t‐SEED is optimal if it has the largest possible dimension. In this paper, we investigate the bounds on the dimensions of 2‐SEEDs systematically. The exact dimensions of optimal 2‐ SEEDs are almost determined, with five possible exceptions in doubt. General upper bounds on dimensions of 2‐ SEEDs are demonstrated, the corresponding leave graphs are described, and several exceptional cases are studied in details. Meanwhile, we employ 2‐homogenous groups to obtain new lower bounds on the dimensions of 2‐ SEEDs for prime power orders v and general block sizes k.