Two-mode binomial code[1] 

Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients.

The simplest two-mode \(S=1\) code is an analogue of the "0-2-4" single-mode binomial code, with codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{2}}\left(|40\rangle+|04\rangle\right)\\ |\overline{1}\rangle&=|22\rangle~. \end{split} \tag*{(1)}\end{align}

The general codewords are \begin{align} |\overline{\mu}\rangle=\frac{1}{2^{J}}\sum_{m=0}^{2J}\left(-1\right)^{\mu m}\sqrt{{2J \choose m}}\left|2J-(S+1)m,(S+1)m\right\rangle~, \tag*{(2)}\end{align} with spacing \(S\) and dephasing error parameter \(N\) such that \(J = \frac{1}{2}(N+1)(S+1)\) [2]. The \(S=0\) version can be obtained by applying a \(50:50\) beamsplitter to the highest-weight Fock state \(|2J,0\rangle\) [3].