A Construction of 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 ${{\mathcal {B}}}$ of k-subsets of a v-set V with a partition ${{\mathcal {B}}}_1,\mathcal{B}_2,\ldots ,{{\mathcal {B}}}_{m}$ of ${{\mathcal {B}}}$ satisfying ${{|\{B\in {\mathcal {B}}_i:\, E \subseteq B\}|}\over {|{\mathcal {B}}_i|}}=\mu _E$ for any $1\le i\le m$ and $E\subseteq V$, $|E|\le t$, where $\mu _E$ 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 $k\mid v$, an optimal 1-(v, k; m) SEED has dimension ${v-1\atopwithdelims ()k-1}$ and can be constructed by Baranyai’s Theorem. This note investigates optimal 1-(v, k; m) SEEDs with $k\not \mid v$, 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 $v\equiv -s$ (mod k), $k\ge s+1$ and $v\ge \max \{2k, s(2k-1)\}$.