A Construction of Optimal 1-Spontaneous Emission Error Designs

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 \(v\ge \max \{2k, s(2k-1)\}\).