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A t-spontaneous emission error design, denoted by t-(v, k; m) SEED or t-SEED in short, is a system B of k-subsets of a v-set V with a partition B 1 , B 2 , … , B m of B satisfying | { B ∈ B i : E ⊆ B } | | B i | = μ E for any 1 ≤ i ≤ m and E ⊆ V , | E | ≤ t , where μ E is a constant depending only on E. A t-(v, k; m) SEED is an important combinatorial object with applications in quantum jump codes. The number m is called the dimension of the t-SEED and this corresponds to the number of orthogonal basis states in a quantum jump code. For given v, k and t, a t-(v, k; m) SEED is called optimal when m achieves the largest possible dimension. When k ∣ v , an optimal 1-(v, k; m) SEED has dimension v - 1 k - 1 and can be constructed by Baranyai’s Theorem. This note investigates optimal 1-(v, k; m) SEEDs with k ∤ v , in which a generalization of Baranyai’s Theorem plays a significant role. To be specific, we construct an optimal 1-(v, k; m) SEED for all positive integers v, k, s with v ≡ - s (mod k), k ≥ s + 1 and v ≥ max { 2 k , s (2 k - 1) } . [ABSTRACT FROM AUTHOR]

Abstract The combinatorial object named t -spontaneous emission error design (t -SEED) was proposed by Beth et al. in 2003 in order to correct errors caused by quantum jumps. The newly rising category of t -SEEDs has been studied extensively in recent years. Especially, the maximal possible dimensions for 2-SEEDs with block size 3 were determined completely; lower bounds on 2-SEEDs were established by applying affine groups. In this paper we utilize the action of twisted affine groups on finite fields and obtain new lower bounds on the dimensions of 2- (q 2 , k ; m) SEEDs, some of which outperform the known ones. [ABSTRACT FROM AUTHOR]

Abstract The combinatorial object named t -spontaneous emission error design (t -SEED) was proposed by Beth et al. in 2003 to correct errors caused by quantum jumps. The newly rising category of t -SEEDs contains the classical problem of large sets as a subbranch at the extremal optimal situation. In this article we study carefully the necessary conditions for the existence of a 3- (v , 4 ; m) SEED where v ≡ 5 (mod 6), v ≥ 23 and m = v − 7 (the largest possible dimension). By considering the derived designs we determine all possible degeneracy graphs to twenty types (including the empty graph). Then we define triad graphs to investigate these degeneracy graphs. Finally we rule out eighteen of them and reduce the leave of the putative optimal 3-SEED to three concrete types. [ABSTRACT FROM AUTHOR]

A t-spontaneous emission error design, denoted by t- [ABSTRACT FROM AUTHOR]

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- [ABSTRACT FROM AUTHOR]

Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3-SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3- [ABSTRACT FROM AUTHOR]