Galois-qudit quantum RM code[1]

Description

True Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [1][2; Sec. 4.2].

The CSS construction yields the code \begin{align} [[q^m,k(v_2)-k(v_1),\min\{d(v_1^{\perp}),d(v_2)\}]]_q \tag*{(1)}\end{align} constructed from the generalized Reed Muller Codes RM\(_q(v_1,m)\) and RM\(_q(v_2,m)\), with \(0\leq v_1 \leq v_2 \leq m(q-1)-1\) [1]. The parameters are \begin{align} k(v) &= \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq} \tag*{(2)}\\ d(v) &= (R+1)q^{Q}~, \tag*{(3)}\end{align} 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 a prime power, and \(m\) is a positive integer.

Using the code GRM\(_{q^2}(v,m)\) for \(0\leq v \leq m(q-1)-1\), the Hermitian construction yields the pure quantum code \begin{align} [[q^{2m},q^{2m}-2k(v),d(v^{\perp})]]_q \tag*{(4)}\end{align} [1], where \begin{align} k(v) &= \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq^2}{v-jq^2} \tag*{(5)}\\ d(v^{\perp}) &= (R+1)q^{2Q}~, \tag*{(6)}\end{align} 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 \begin{align} P(C) = \{(a_ib_i)_{i=1}^{n} \mid a \in C_1, b \in C_2^{\perp}\}^{\perp}~. \tag*{(7)}\end{align} Quantum RM codes can be punctured to any length \(r\), provided \begin{align} P(C) = \mathcal{R}_q(v_2-v_1,m) \tag*{(8)}\end{align} has a codeword of this weight. Likewise, the Hermitian puncture code contains \(\mathcal{R}_{q^2}(\mu,m)^\perp|_{\mathbb{F}_q}\) for \((q+1)\nu \leq \mu \leq m(q^2-1)-1\), yielding punctured descendants with distance at least that of the parent code [1].