Prime-qudit RS code [1]

Description

Prime-qudit CSS code constructed using two RS codes.

A 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 \(\mathbb{F}_p=\mathbb{Z}_p\). 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 entry is \(f_{\mu\cup c}\) evaluated at \(\alpha_i\), 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}

Logical states can instead be labeled by the \(k\) highest-degree coefficients, with the superposition running over the \(g+1-k\) lowest-degree ones. Inverting the evaluation points and applying single-qudit multiplication gates maps the code above onto a code of this form with the same parameters. This second form also allows zero as an evaluation point, so that \(p\geq n\). The case \(n=p\), for which the evaluation points exhaust \(\mathbb{Z}_p\), is constructed from extended GRS codes.

A related qubit construction [3] expands a \([N,K,\delta]_{2^k}\) RS code with \(N=2^k-1\), \(K=N-\delta+1\), and \(\delta>N/2+1\) in a self-dual basis of \(\mathbb{F}_{2^k}\) over \(\mathbb{F}_2\). This yields an \([[kN,k(N-2K),d\geq K+1]]\) qubit code. These qubit codes are binarizations of Galois-qudit RS codes rather than members of this family.