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Bounds on the Dimensions of 2‐Spontaneous Emission Error Designs
Author(s) -
Zhou Junling,
Chang Yanxun
Publication year - 2016
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21514
Subject(s) - mathematics , dimension (graph theory) , quantum , prime power , jump , combinatorics , prime (order theory) , upper and lower bounds , discrete mathematics , code (set theory) , basis (linear algebra) , mathematical analysis , quantum mechanics , set (abstract data type) , physics , computer science , geometry , programming language
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‐ ( v , 3 ; m ) SEEDs are almost determined, with five possible exceptions in doubt. General upper bounds on dimensions of 2‐ ( v , 4 ; m ) 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‐ ( v , k ; m ) SEEDs for prime power orders v and general block sizes k .