Galois-qudit quantum RM code

Galois-qudit quantum RM code [1]

Description

True Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes, their punctured or shortened versions, or projective RM codes, via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [1,3][2; Sec. 4.2][4; Sec. 6.2].

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 $0\leq R<q-1$. Here $0\leq v_1,v_2 \leq m(q-1)-1$, $q$ is a prime power, and $m$ is a positive integer. For $0\leq v_1\leq v_2<n\leq q$, evaluating at only $n$ of the points for $m=1$ yields $[[n,v_2-v_1,\min(v_1+2,n-v_2)]]_q$ codes.

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 $v+1 = (q^2 - 1)Q + R$.

The quantum code itself can also be punctured. For a CSS code constructed from classical codes $C_1$ and $C_2$, the puncture 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 $(q+1)v \leq \mu \leq m(q^2-1)-1$, yielding punctured descendants with distance at least that of the parent code [1].

Protection

The CSS family is pure with distance $\min\{d(v_1^\perp),d(v_2)\}$, while the Hermitian family is pure with distance $d(v^\perp)$, both for GRM codes evaluated at all points [1]. Punctured descendants of the quantum code retain at least the parent distance [1]. QRM$_{d}(m)$ quantum codes are $\mathcal{M}_{d}^{m}$ distillation codes of distance $D=2$ [5]. We define a $\mathcal{M}_{d}^{m}$ distillation code as any stabilizer code $C$ on $n$ Galois qudits having the following properties: (a) All $M \in \mathcal{M}_{d}^{m}$ are transversal so that $M^{\otimes n}C(M^{\otimes n})^{\dagger} = M_{L}^{\dagger}CM_{L}$, (b) the code has distance $D \geq 2$, and (c) the code has logical Pauli operators $X_{L} = X[\mathbf{1}]$ and $Z_{L} = Z[(d-1)\mathbf{1}]$. Here, $\mathbf{1}$ is a shorthand for the vector $(1,1, \dots, 1)$.

Rate

The CSS family has rate $ (k(v_2)-k(v_1) )/q^m$, while the Hermitian family has rate $1-2k(v)/q^{2m}$ [1].

Cousins

Generalized RM (GRM) code— Generalized RM codes can be used to construct Galois-qudit RM codes via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [1][2; Sec. 4.2].
Projective RM (PRM) code— Projective RM codes can be used to construct Galois-qudit RM codes [3][4; Sec. 6.2].
Galois-qudit CSS code— Galois-qudit RM codes admit a CSS subfamily built from nested GRM codes $ \mathrm{GRM}_q(v_1,m) \subseteq \mathrm{GRM}_q(v_2,m) $ [1].
Hermitian Galois-qudit code— Galois-qudit RM codes admit a Hermitian subfamily built from $\mathrm{GRM}_{q^2}(v,m)$ codes contained in their Hermitian duals [1].
Quantum maximum-distance-separable (MDS) code— Quantum GRM codes yield quantum MDS families $[[q,q-2\nu-2,\nu+2]]_q$ for $0 \leq \nu \leq (q-2)/2$, $[[q^2,q^2-2\nu-2,\nu+2]]_q$ for $0 \leq \nu \leq q-2$, and punctured descendants $[[(\nu+1)q,(\nu+1)q-2\nu-2,\nu+2]]_q$ for $0 \leq \nu \leq q-2$ [1].

Member of code lists

Primary Hierarchy

Quantum Domain

Galois-qudit Kingdom

Galois-qudit codeQECC Quantum

Galois-qudit USt code

Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

True Galois-qudit stabilizer code

Parents

True Galois-qudit stabilizer code

Galois-qudit RM codes can be constructed via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [1][2; Sec. 4.2].

Galois-qudit quantum RM code

Children

Prime-qudit RM codeQuantum RM code

Prime-qudit RM codes are the Galois-qudit RM codes with prime $q$ that are constructed via the CSS construction from GRM codes, their punctured or shortened versions, or their duals.

Galois-qudit RS code

Galois-qudit RS codes are the Galois-qudit RM codes from the CSS construction with $m=1$ whose two GRM codes are punctured to the same $n$ evaluation points. For $m=1$, GRM codes are extended RS codes [1]. Every Galois-qudit RS code takes this form when its logical states are labeled by the highest-degree coefficients, possibly after inverting the evaluation points and applying single-Galois-qudit multiplication gates.

References

[1]
P. K. Sarvepalli and A. Klappenecker, “Nonbinary Quantum Reed-Muller Codes”, (2005) arXiv:quant-ph/0502001
[2]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[3]
“Nonbinary Stabilizer Codes”, Mathematics of Quantum Computation and Quantum Technology 305 (2007) DOI
[4]
S. A. Aly, “On Quantum and Classical Error Control Codes: Constructions and Applications”, (2008) arXiv:0812.5104
[5]
E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) arXiv:1205.3104 DOI

Page edit log

Victor V. Albert (2026-09-23) — most recent
Victor V. Albert (2026-09-22)
Victor V. Albert (2026-06-08)
Victor V. Albert (2023-11-22)
Shuubham Ojha (2023-11-22)

Cite as:

“Galois-qudit quantum RM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/galois_reed_muller, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/evaluation/galois_reed_muller.yml.