Prime-qudit RS code[1]

Description

Prime-qudit CSS code constructed using two RS codes.

The original construction [1] was for a qubit code (\(p=2\)) by using a basis for a larger Galois field over \(GF(2)\), yielding an \([[kN,k(N-2K),K+1]]\) qubit code from a \([N,K,\delta]_{GF(2^k)}\) RS code with \(N=2^k-1\) and \(K=N-\delta+1\).

An alternative construction [2] yields an \([[n,k,d]]_{p>n}\) prime-qudit CSS code with \(d=\min(n-g,g+2-k)\) that is constructed using two RS codes over \(GF(p)=\mathbb{Z}_p\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(\mathbb{Z}_p\), and let \(g\) be a number satisfying \(0\leq k \leq g < n\). Then, define degree-\(g\) polynomials \begin{align} f_{\mu\cup c}\left(x\right)=\mu_{0}+\mu_{1}x+\cdots+\mu_{k-1}x^{k-1}+c_{k}x^{k}+\cdots+c_{g}x^{g}\,, \tag*{(1)}\end{align} where the first \(k\) coefficients are indexed by the coefficient vector \(\mu\in\mathbb{Z}_p^{ k}\), and the remaining coefficients are indexed by the vector \(c\in\mathbb{Z}_p^{ (g+1-k)}\). Logical states, labeled by \(\mu\), are superpositions of canonical basis states whose \(i\)th bit is \(f_{\mu\cup c}\), evaluated at \(\alpha_i\) and summed over all possible vectors \(c\), \begin{align} |\overline{\mu}\rangle=\sum_{c\in\mathbb{Z}_{p}^{(g+1-k)}}|f_{\mu\cup c}(\alpha_{1}),|f_{\mu\cup c}(\alpha_{2}),\cdots,|f_{\mu\cup c}(\alpha_{n})\rangle. \tag*{(2)}\end{align}