Optimal 1—Spontaneous Emission Error Designs*

Abstract

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. The design of t-SEED was introduced by Beth et al. in 2003 (T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, M. Mussinger, Des Codes Cryptogr 29 (2003), 51-70) to construct quantum jump codes. The number m of designs in a t-(v, k; m) SEED is called dimension, which corresponds to the number of orthogonal basis states in a quantum jump code. A t-SEED is nondegenerate if every point appears in each of its member design.A nondegenerate t-SEED is called optimal when it achieves the largest possible dimension. This paper investigates the dimension of optimal 1-SEEDs, in which Baranyai's Lemma plays a significant role and the hypergraph distribution is closely related as well. Several classes of optimal 1-SEEDs are shown to exist. In particular, we determine the exact dimensions of optimal 1-(v, k; m) SEEDs for all orders v and block sizes k with 2 ≤ k ≤ 6.Let v, k, t, m be integers with 0 < t < k < v and V be a set of v elements (or points)where μ E is a constant depending only on E. The combinatorial structure t-SEED is employed to construct quantum jump codes by [3]. In this application, the number m of designs in a t-(v, k; m) SEED corresponds to the dimension (number of orthogonal basis states) of a quantum jump code and it is required to be as large as possible. Following this, we also call the parameter m of a t-(v, k; m) SEED its dimension. Denote by M (t, k, v) the largest possible dimension m for which a t-(v, k; m) SEED exists. Then we have the following basic result, which is presented in [3] in terms of quantum jump codes.A t-SEED can be viewed as a collection of mutually disjoint t-packings. A t- (v, k, λ) packing, is a pair (X, B), where V is a v-set of points and B is a collection of k-subsets of V (called blocks), with the property that every t-subset of V occurs in at most λ blocks. The parameter λ is the index of the packing and, in a t- (v, k, λ) packing, a t-subset contained in λ blocks is always assumed to exist. In particular, if every t-subset of V is contained in exactly λ blocks, then the packing becomes a t- (v, k, λ) design. A t-(v, k, λ) design is also denoted by S λ (t, k, v), dropping the subscript when λ = 1.Particularly, m mutually disjoint t- (v, k, λ) designs yields a t-(v, k; m) SEED. A large set of t-designs, denoted by LS λ (t, k, v), is a partition of all k-subsets of V into N mutually disjoint t- (v, k, λ) designs, where N = v−t k−t /λ. By convention, an LS(t, k, v) denotes an LS λ (t, k, v) with λ = 1. Lin and Jimbo [6] showed that a t-(v, k; m) SEED attaining the upper bound in Lemma 1.1 is equivalent to a large set of Steiner t-designs LS(t, k, v). Lemma 1.2. ([ 6, Corollary 1]) A t-(v, k; m) SEED attaining the upper bound in Lemma 1.1 exists if and only if a large set...