Galois-qudit RS code[1]

Also called a polynomial code (QPyC). An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two Reed-Solomon codes over \(GF(q)\).

Let \(C_1\) be a \([n,k_1,d_1]_q\) Reed-Solomon code and \(C_2^\perp\) be a \([n,k_2,d_2]_q\) Reed-Solomon code, modified such that \(C_2^\perp \subseteq C_1\) and \(0\le k_2 \le k_1 \le n\). Then, a polynomial code is a non-degenerate \([[n,k_2,d]]_q\) Galois-qudit CSS code with \(d=\min(n-k_1+1,k_1-k_2+1)\). The polynomial code is the span of the basis codewords over GF(\(q\)) \begin{align} |\overline{\beta_0,\cdots,\beta_{k_2-1}}\rangle = \sum_{(\beta_{k_2},\cdots,\beta_{k_1-1})\in GF(q) } \bigotimes_{i=1}^{n} \left| \sum_{j=0}^{k_1-1} \beta_j \alpha_i^j \right\rangle, \tag*{(1)}\end{align} where \((\alpha_1, \cdots, \alpha_n)\) are \(n\) distinct points chosen for code \(C_1\) from \(GF(q)\setminus \{0\}\).'