Galois-qudit quantum RM code[1] 

True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [2; Sec. 4.2].

The CSS construction yields a \([[q^m,k(v_2)-k(v_1),\min\{d(v_1^{\perp}),d(v_2)\}]]_q\) code, constructed from the generalized Reed Muller Codes RM\(_q(v_1,m)\) and RM\(_q(v_2,m)\), with \(0\leq v_1 \leq v_2 \leq q(m-1)-1\). The parameters are \(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}\) and \(d(v) = (R+1)q^{Q}\), where \(m(q-1)-v=(q-1)Q+R\) so that \(0\leq R\leq q-1\). Here \(0\leq v_1,v_2 \leq m(q-1)-1\), \(q\) is prime, and \(m\) is a positive integer.

Using the code GRM\(_{q^2}(v,m)\) for \(0\leq v \leq q(m-1)-1\), the Hermitian construction yields a pure \([[q^{2m},q^{2m}-k(v),d(v^{\perp})]]_q\) quantum code where \(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq^2}{v-jq^2}\) and \(d(v^{\perp}) = (R+1)q^{2Q}\) with \(v+1 = (q^2 - 1)Q + R\).

For a CSS code constructed from classical codes \(C_1\) and \(C_2\), the punctured code is defined as the classical code \(P(C) = \{(a_ib_i)_{i=1}^{n}|a \in C_1, b \in C_2^{\perp}\}^{\perp}\). Quantum RM codes can be punctured to any length \(r\), provided \(P(C) = \mathcal{R}_q(v_2-v_1,m)\) has a codeword of this weight.