---
title: Truncated MacWilliams LP System
url: https://www.emergentmind.com/topics/truncated-macwilliams-linear-programming-system
type: topic
---

# Truncated MacWilliams LP System

Searching arXiv for the cited papers and closely related work on MacWilliams transforms and LP bounds.
A truncated MacWilliams linear-programming system is a relaxation of the full MacWilliams feasibility system in which only a selected subset of transform sectors is retained explicitly, while omitted sectors are handled by positivity, normalization, and slack-variable bounds. In the intrinsic framework for quantum codes, the system is built from the decomposition of the conjugation representation on $\mathcal{L}(V)$, from quadratic projector and twirl enumerators, and from the intrinsic MacWilliams transform relating them; in multiplicity-free settings this yields a linear program, whereas multiplicities lead naturally to semidefinite feasibility problems [2604.16023]. For permutation-invariant qudit codes, the same construction specializes to symmetric-power representations of $\mathrm{SU}(q)$, where the MacWilliams matrix is an explicit finite Racah transform and truncation provides computable upper bounds on admissible code parameters [2605.15372].

## 1. Formal setting and basic objects

An intrinsic quantum code is a subspace $C \subset V$, where $V$ is a finite-dimensional unitary representation of a group $G$. If $\Pi_C$ denotes the orthogonal projector onto $C$, code constraints are expressed in the operator space $\mathcal{L}(V)$ equipped with the Hilbert–Schmidt inner product and the conjugation action
\[
g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).
\]
The operator space decomposes isotypically as
\[
\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,
\]
with $\dim W_\xi=d_\xi m_\xi$ when the irreducible $\xi$ has dimension $d_\xi$ and multiplicity $m_\xi$. In the multiplicity-free case, $m_\xi \in \{0,1\}$ and the summands are orthogonal irreducible sectors [2604.16023].

For $\mathrm{SU}(2)$ with $V \cong V_j$ the spin-$j$ irrep, one has
\[
\mathcal{L}(V)\cong V_j \otimes V_j^* \cong \bigoplus_{k=0}^{2j} V_k,
\]
where $V_k$ is the $(2k+1)$-dimensional space of rank-$k$ irreducible tensor operators. For permutation-invariant qudit codes, the physical space is
\[
V_n := \mathrm{Sym}^n(\mathbb{C}^q),
\]
the $n$th symmetric power of the defining representation of $\mathrm{SU}(q)$, with
\[
\dim V_n=\binom{n+q-1}{n}.
\]
Under conjugation,
\[
\mathcal{L}(V_n)\cong \bigoplus_{a=0}^n W_a, \qquad W_a\cong (a,0,\dots,0,a),
\]
again multiplicity-free, with sector dimensions
\[
d_a=\frac{2a+q-1}{q-1}\,\binom{a+q-2}{q-2}^{2}.
\]
This multiplicity-free structure is the algebraic condition that makes a scalar MacWilliams matrix and an LP formulation available [2605.15372].

The truncated system preserves this representation-theoretic organization but restricts explicit constraints to a chosen index set, typically the low-degree or low-spin sectors most relevant to the target detection requirement. This suggests that truncation is not an ad hoc numerical shortcut, but a structured relaxation of the full intrinsic feasibility problem.

## 2. Projector and twirl enumerators

Fix an orthogonal decomposition
\[
\mathcal{L}(V)=\bigoplus_{i \in I} W_i
\]
with orthogonal projectors $P_i:\mathcal{L}(V)\to \mathcal{L}(V)$. If $B_i$ is an orthonormal basis of $W_i$, then
\[
P_i(X)=\sum_{E \in B_i} \langle E,X\rangle E.
\]
The associated projector sesquilinear form is
\[
A_i(X_1,X_2):=\langle X_1,P_i(X_2)\rangle
 =\sum_{E \in B_i}\mathrm{Tr}(X_1^\dagger E)\mathrm{Tr}(X_2 E^\dagger),
\]
with quadratic form
\[
A_i(X,X)=\|P_i(X)\|_2^2 \ge 0.
\]

The twirling superoperator is
\[
T_i(X):=\sum_{E \in B_i} E^\dagger X E,
\]
independent of the orthonormal basis choice. Its sesquilinear form is
\[
B_i(X_1,X_2):=\langle X_1,T_i(X_2)\rangle
=\sum_{E \in B_i}\mathrm{Tr}(X_1^\dagger E^\dagger X_2 E),
\]
with
\[
B_i(X,X)=\sum_{E \in B_i}\mathrm{Tr}(X^\dagger E^\dagger X E).
\]

These two families satisfy positivity and normalization:
\[
A_i(X,X)\ge 0,\qquad \sum_i A_i(X,X)=\mathrm{Tr}(X^\dagger X),
\]
and
\[
\sum_i B_i(X,X)=|\mathrm{Tr}(X)|^2.
\]
If $W_0=\mathrm{span}\{I\}$ and $B_0=\{I/\sqrt{|V|}\}$, then
\[
A_0(X,X)=\frac{|\mathrm{Tr}(X)|^2}{|V|}, \qquad
B_0(X,X)=\frac{\mathrm{Tr}(X^\dagger X)}{|V|}.
\]

For positive semidefinite $X$ there is a Knill–Laflamme type inequality,
\[
A_i(X,X)\le \mathrm{rank}(X)\, B_i(X,X),
\]
and for a code projector $P$ of rank $K$,
\[
A_i(P,P)\le K\,B_i(P,P).
\]
Equality holds exactly when the corresponding sector is detected by the code. In the intrinsic framework, “distance” or “depth” means detection of all sectors in a prescribed error set rather than physical tensor weight. For symmetric-power representations of $\mathrm{SU}(q)$, intrinsic depth equals ordinary qudit distance when codes are restricted to the permutation-invariant subspace [2604.16023].

In the permutation-invariant qudit specialization, the LP can also be expressed through sector moments
\[
x_a := \mathrm{Tr}(P_a X)\ge 0,\qquad
m_b := \mathrm{Tr}(T_b(X))=\sum_{a=0}^n M_{ba}x_a,
\]
with $\sum_a x_a=K$ for a rank-$K$ projector $\Pi$. These are the primal and MacWilliams moments used in the finite Racah-transform formulation [2605.15372].

## 3. Intrinsic MacWilliams transform

When the conjugation representation on $\mathcal{L}(V)$ is multiplicity-free, Schur’s lemma makes the intertwiner algebra commutative, and the normalized families
\[
\{d_\xi^{-1/2}P_\xi\}_\xi
\quad\text{and}\quad
\{d_\xi^{-1/2}T_\xi\}_\xi
\]
form orthonormal bases of $\mathrm{Hom}_G(\mathcal{L}(V),\mathcal{L}(V))$. There is a unique unitary $U$ such that
\[
d_\xi^{-1/2}T_\xi = \sum_\rho U_{\xi\rho}\, d_\rho^{-1/2}P_\rho.
\]
Writing
\[
M_{\xi\rho}:=\sqrt{\frac{d_\xi}{d_\rho}}\,U_{\xi\rho},
\]
one obtains the intrinsic MacWilliams identity
\[
B_\xi(P,P)=\sum_\rho M_{\xi\rho}\,A_\rho(P,P),
\]
or in vector form,
\[
B=MA.
\]
The transform satisfies the weighted orthogonality relation
\[
M D \overline{M}^{\,T}=D,\qquad D:=\mathrm{diag}(d_\xi).
\]
In Plancherel-normalized variables, the transform is unitary [2604.16023].

For $\mathrm{SU}(2)$, the transform admits a closed form in terms of Wigner $6j$-symbols:
\[
[U]_{k_1k_2}
=
(-1)^{2j+k_1+k_2}
\sqrt{(2k_1+1)(2k_2+1)}
\begin{Bmatrix}
j & j & k_1 \\
j & j & k_2
\end{Bmatrix},
\]
and the unnormalized matrix is
\[
M_{k_1k_2}
=
(-1)^{2j+k_1+k_2}(2k_1+1)
\begin{Bmatrix}
j & j & k_1 \\
j & j & k_2
\end{Bmatrix}.
\]
The transform is the Racah transform between two coupling schemes in $(V_j^{\otimes 4})^{\mathrm{SU}(2)}$ [2604.16023].

For permutation-invariant qudit codes, the MacWilliams matrix is an explicit finite Racah transform:
\[
M_{ba}
=
\frac{d_b}{\dim V_n}
\;{}_4F_3\!\left(
\begin{matrix}
-b,\ b+q-1,\ -a,\ a+q-1\\
q-1,\ -n,\ n+q
\end{matrix}
;1\right),
\qquad a,b=0,\dots,n.
\]
Special cases are
\[
M_{0a}=\frac{1}{\dim V_n},\qquad M_{b0}=\frac{d_b}{\dim V_n}.
\]
The matrix satisfies
\[
MDM^T=D,\qquad d_aM_{ba}=d_bM_{ab},\qquad M^2=I.
\]
Its rows are Racah orthogonal polynomials on the quadratic lattice
\[
y_a=a(a+q-1),
\]
and the eigenvalues of the degree-one twirl lie on the affine image
\[
x_a=A-B\,y_a
\]
with
\[
A=\frac{q^2-1}{\dim V_n},\qquad
B=\frac{q(q^2-1)}{\dim V_n\,n(q-1)(n+q)}.
\]
The tridiagonal multiplication rule
\[
T_1T_b
=
N_{1b}^{b+1}T_{b+1}
+
N_{1b}^{b}T_b
+
N_{1b}^{b-1}T_{b-1}
\]
yields a stable three-term recurrence for constructing transform rows [2605.15372].

## 4. Linear-programming formulation and truncation

In the multiplicity-free intrinsic setting, let
\[
A_\xi:=A_\xi(P,P),\qquad B_\xi:=B_\xi(P,P),
\]
for a code projector $P$, and let $K:=\dim C$, $N:=\dim V$. The full feasibility system consists of
\[
A_\xi \ge 0,\qquad A_\xi \le K B_\xi,\qquad \sum_\xi A_\xi=K,\qquad \sum_\xi B_\xi=K^2,\qquad B=MA,
\]
together with detection equalities
\[
A_\xi=K B_\xi \qquad (\xi \in \mathcal{E})
\]
for the chosen detected sector set $\mathcal{E}$ [2604.16023].

In normalized variables
\[
\widetilde A_\xi:=\frac{N}{K^2}A_\xi,\qquad
\widetilde B_\xi:=\frac{N}{K}B_\xi,
\]
one has
\[
\widetilde A_0=\widetilde B_0=1,\qquad
\widetilde B=K M \widetilde A,
\]
with positivity $\widetilde A_\xi\ge 0$, dominance $\widetilde A_\xi\le \widetilde B_\xi$, and detection equalities $\widetilde A_\xi=\widetilde B_\xi$ on detected sectors.

Truncation retains only a subset $I_{\mathrm{trunc}}\subset I$ of sectors. In the intrinsic quantum formulation, one introduces slack variables
\[
S_A:=\sum_{\ell \notin I_{\mathrm{trunc}}}\widetilde A_\ell,\qquad
S_B:=\sum_{\ell \notin I_{\mathrm{trunc}}}\widetilde B_\ell,
\]
subject to
\[
S_A\ge 0,\qquad S_A\le S_B,
\]
and the truncated normalizations
\[
\sum_{k \in I_{\mathrm{trunc}}}\widetilde A_k + S_A = \frac{N}{K},\qquad
\sum_{k \in I_{\mathrm{trunc}}}\widetilde B_k + S_B = KN.
\]
For retained sectors,
\[
\widetilde B_k
=
K \sum_{\ell \in I_{\mathrm{trunc}}} M_{k\ell}\widetilde A_\ell
+
K \sum_{\ell \notin I_{\mathrm{trunc}}} M_{k\ell}\widetilde A_\ell.
\]
The omitted contribution can be bounded linearly using nonnegativity and bounds on $S_A$. This relaxation preserves validity of upper bounds because the feasible region enlarges, although tightness may be lost [2604.16023].

For the explicit truncated $\mathrm{SU}(2)$ system with $V_j \cong \mathrm{Sym}^{2j}(\mathbb{C}^2)$ and indices $k=0,1,\dots,R\le 2j$, the retained variables are $\{\widetilde A_k,\widetilde B_k\}_{k=0}^R$ and slacks $(S_A,S_B)$, with constraints
\[
\widetilde A_k\ge 0,\qquad \widetilde A_k\le \widetilde B_k,
\]
\[
\widetilde B_k
=
K \sum_{\ell=0}^R M_{k\ell}\widetilde A_\ell + K t_k,
\]
where $t_k$ absorbs omitted terms and may be bounded as
\[
t_k \in [m_k^- S_A,\, m_k^+ S_A]
\]
if row-wise bounds are known. Detection equalities are imposed for $k\in\{1,\dots,d-1\}$, and choosing $R$ close to $d$ is reported to yield good bounds with small LPs [2604.16023].

The permutation-invariant qudit version uses variables $x_a\ge 0$ with $\sum_a x_a=K$ and dual variables
\[
m_b=\sum_{a=0}^n M_{ba}x_a\ge 0.
\]
A truncated LP enforces only the constraints for $b=0,1,\dots,T$. This is a safe relaxation because fewer dual inequalities are imposed, while each retained row still uses the exact finite transform. The three-term recurrence makes the retained rows numerically stable to compute [2605.15372].

## 5. Explicit systems and extremality certificates

The intrinsic LP framework yields exact feasibility systems for specific permutation-invariant codes, and in the examples treated the feasible point is unique. The resulting uniqueness certifies extremality statements about code dimension or distance [2604.16023].

| Example | Parameters | Unique feasible solution and certification |
|---|---|---|
| Four-qubit | $\mathrm{SU}(2)$, $j=2$, $N=5$, $K=2$, detection set $\mathcal{E}=\{1\}$ | $\widetilde A=(1,0,0,0,3/2)$, $\widetilde B=(1,0,20/7,5/2,51/14)$; certifies extremality of the permutation-invariant code $((4,2,2))_2$ |
| Seven-qubit | $\mathrm{SU}(2)$, $j=7/2$, $N=8$, $K=2$, detection set $\mathcal{E}=\{1,2\}$ | $\widetilde A=(1,0,0,0,0,0,3,0)$, $\widetilde B=(1,0,0,49/11,0,49/13,3,540/143)$; certifies extremality of $((7,2,3))_2$ |
| Three-qutrit | $\mathrm{SU}(3)$, $\mathrm{Sym}^3(\mathbb{C}^3)$, $N=10$, $K=2$, detection set $\mathcal{E}=\{(1,1)\}$ | $\widetilde A=(1,0,0,4)$, $\widetilde B=(1,0,45/7,88/7)$; certifies extremality of the permutation-invariant code $((3,2,2))_3$ |

In the four-qubit case, the constraints are
\[
\widetilde A_0=\widetilde B_0=1,\qquad
\widetilde B=2M\widetilde A,\qquad
\widetilde A_k\ge 0,\qquad
\widetilde A_k\le \widetilde B_k,\qquad
\widetilde A_1=\widetilde B_1,
\]
with
\[
\sum_{k=0}^4 \widetilde A_k=\frac{5}{2},\qquad
\sum_{k=0}^4 \widetilde B_k=10.
\]
The unique feasible solution certifies that there is no $((n,2,2))_2$ for $n<4$, no $((4,K,2))_2$ with $K>2$, and no $((4,2,d))_2$ with $d>2$ [2604.16023].

In the seven-qubit case, the normalized system is
\[
\widetilde A_0=\widetilde B_0=1,\qquad
\widetilde B=2M\widetilde A,
\]
with $\widetilde A_1=\widetilde B_1$, $\widetilde A_2=\widetilde B_2$,
\[
\sum_{k=0}^7 \widetilde A_k=4,\qquad
\sum_{k=0}^7 \widetilde B_k=16.
\]
The unique feasible solution yields minimality of $n=7$ for $d=3$ with $K=2$, and excludes both $((7,2,d))_2$ with $d>3$ and $((7,K,3))_2$ with $K>2$ [2604.16023].

The qudit paper also gives a worked truncated example for $q=2$, $n=3$, with $N=4$, sector dimensions $d_a=2a+1$, and truncation level $T=2$. The retained constraints are
\[
m_0=\frac{K}{4}\ge 0,
\]
\[
m_1=\tfrac{3}{4}x_0+\tfrac{11}{20}x_1+\tfrac{3}{20}x_2-\tfrac{9}{20}x_3\ge 0,
\]
\[
m_2=\tfrac{5}{4}x_0+\tfrac{1}{4}x_1-\tfrac{3}{4}x_2+\tfrac{1}{4}x_3\ge 0,
\]
with an example detection condition $m_1=0$ [2605.15372].

## 6. Multiplicity, semidefinite extension, and broader variants

When multiplicities occur, the intrinsic MacWilliams system ceases to be scalar. One fixes a decomposition
\[
W_\xi=\bigoplus_{\alpha=1}^{m_\xi} W_{\xi\alpha},\qquad W_{\xi\alpha}\cong \xi,
\]
together with coherent isometric intertwiners $\phi^\xi_{\alpha\beta}:W_{\xi\alpha}\to W_{\xi\beta}$. The corresponding projector matrix units $\Pi_{\xi\alpha\beta}$ and twirl matrix units $\mathrm{Twirl}_{\xi\alpha\beta}$ form orthogonal bases of the intertwiner algebra. The MacWilliams transform becomes block unitary, and the enumerators become matrix-valued:
\[
[A_\xi(X)]_{\alpha\beta}
=
\sum_{i=1}^{d_\xi}\mathrm{Tr}(X^\dagger E_i^\beta)\mathrm{Tr}(X(E_i^\alpha)^\dagger),
\]
\[
[B_\xi(X)]_{\alpha\beta}
=
\sum_{i=1}^{d_\xi}\mathrm{Tr}(X^\dagger (E_i^\alpha)^\dagger X E_i^\beta).
\]
These matrices are Hermitian PSD blockwise and satisfy
\[
\sum_\xi \mathrm{Tr}\,A_\xi(X)=\mathrm{Tr}(X^\dagger X),\qquad
\sum_\xi \mathrm{Tr}\,B_\xi(X)=|\mathrm{Tr}(X)|^2.
\]
If $X\succeq 0$ and $r=\mathrm{rank}(X)$, then
\[
A_\xi(X)\preceq r\,B_\xi(X),
\]
and for a code projector $P$ of rank $K$,
\[
A_\xi(P)\preceq K\,B_\xi(P).
\]
The block MacWilliams identity is
\[
\mathrm{vec}(B)=M\,\mathrm{vec}(A),
\]
so the feasibility problem is an SDP rather than an LP [2604.16023].

The intrinsic SDP constraints are
\[
A_\xi \succeq 0,\qquad A_\xi \preceq K B_\xi,\qquad
\sum_\xi \mathrm{Tr}\,A_\xi=K,\qquad
\sum_\xi \mathrm{Tr}\,B_\xi=K^2,
\]
together with
\[
\mathrm{vec}(B)=M\,\mathrm{vec}(A),\qquad
A_\xi=K B_\xi \quad (\xi\in\mathcal E).
\]
In the first non-multiplicity-free $\mathrm{SU}(3)$ example, with $V\cong (2,2)$ of dimension $27$, the decomposition of $\mathcal{L}(V)$ includes multiplicities such as $2(1,1)$, $3(2,2)$, $2(4,1)$, $2(1,4)$, and $2(3,3)$. For the $5$-dimensional intrinsic code $I:=\chi_6\subset V$ under restriction to the subgroup $3.A_6\subset \mathrm{SU}(3)$, one obtains an SDP upper bound $K\le 5$ among depth-$2$ intrinsic codes, and the code attains the bound [2604.16023].

A broader classical analogue appears in refined MacWilliams-type systems for the Lee, homogeneous, and subfield metrics. There, classical weight partitions are replaced by finer partitions that restore Fourier-reflexive transforms, and LP bounds are derived from the resulting transform inequalities. Truncation is implemented either by keeping only selected dual rows or by retaining only selected primal classes and aggregating the rest into slack variables. In that setting, dropping rows or using lower bounds on omitted contributions remains a valid relaxation and therefore still yields upper bounds [2409.11926].

The relation to classical Delsarte theory is direct: a nonnegative inner distribution, a linear MacWilliams transform, positivity of the dual distribution, and distance or detection constraints together produce the optimization problem. In the intrinsic permutation-invariant setting, the scalar LP is available because the intertwiner algebra is commutative; outside multiplicity-free regimes, the same logic survives only in matrix form, forcing an invariant-SDP generalization [2604.16023].

Source: https://www.emergentmind.com/topics/truncated-macwilliams-linear-programming-system