Truncated MacWilliams LP System

• The paper introduces a truncated MacWilliams LP system that relaxes the full feasibility constraints by explicitly retaining a selected subset of transform sectors while managing others via positivity and slack-variable bounds.
• The methodology leverages intrinsic quantum code decompositions and explicit Racah transforms to convert complex operator constraints into a computable linear program, yielding practical upper bounds on code parameters.
• This approach not only simplifies the computational process for permutation-invariant qudit codes but also extends to semidefinite formulations in cases of multiplicity, thereby certifying extremality conditions.

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 (Kubischta et al., 17 Apr 2026). 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 (Teixeira, 14 May 2026).

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 $\mathrm{SU}(q)$0 when the irreducible $\mathrm{SU}(q)$1 has dimension $\mathrm{SU}(q)$2 and multiplicity $\mathrm{SU}(q)$3. In the multiplicity-free case, $\mathrm{SU}(q)$4 and the summands are orthogonal irreducible sectors (Kubischta et al., 17 Apr 2026).

For $\mathrm{SU}(q)$5 with $\mathrm{SU}(q)$6 the spin-$\mathrm{SU}(q)$7 irrep, one has

$\mathrm{SU}(q)$8

where $\mathrm{SU}(q)$9 is the $C \subset V$0-dimensional space of rank-$C \subset V$1 irreducible tensor operators. For permutation-invariant qudit codes, the physical space is

$C \subset V$2

the $C \subset V$3th symmetric power of the defining representation of $C \subset V$4, with

$C \subset V$5

Under conjugation,

$C \subset V$6

again multiplicity-free, with sector dimensions

$C \subset V$7

This multiplicity-free structure is the algebraic condition that makes a scalar MacWilliams matrix and an LP formulation available (Teixeira, 14 May 2026).

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

$C \subset V$8

with orthogonal projectors $C \subset V$9. If $V$0 is an orthonormal basis of $V$1, then

$V$2

The associated projector sesquilinear form is

$V$3

with quadratic form

$V$4

The twirling superoperator is

$V$5

independent of the orthonormal basis choice. Its sesquilinear form is

$V$6

with

$V$7

These two families satisfy positivity and normalization: $V$8 and

$V$9

If $G$0 and $G$1, then

$G$2

For positive semidefinite $G$3 there is a Knill–Laflamme type inequality,

$G$4

and for a code projector $G$5 of rank $G$6,

$G$7

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 $G$8, intrinsic depth equals ordinary qudit distance when codes are restricted to the permutation-invariant subspace (Kubischta et al., 17 Apr 2026).

In the permutation-invariant qudit specialization, the LP can also be expressed through sector moments

$G$9

with $\Pi_C$0 for a rank-$\Pi_C$1 projector $\Pi_C$2. These are the primal and MacWilliams moments used in the finite Racah-transform formulation (Teixeira, 14 May 2026).

3. Intrinsic MacWilliams transform

When the conjugation representation on $\Pi_C$3 is multiplicity-free, Schur’s lemma makes the intertwiner algebra commutative, and the normalized families

$\Pi_C$4

form orthonormal bases of $\Pi_C$5. There is a unique unitary $\Pi_C$6 such that

$\Pi_C$7

Writing

$\Pi_C$8

one obtains the intrinsic MacWilliams identity

$\Pi_C$9

or in vector form,

$C$0

The transform satisfies the weighted orthogonality relation

$C$1

In Plancherel-normalized variables, the transform is unitary (Kubischta et al., 17 Apr 2026).

For $C$2, the transform admits a closed form in terms of Wigner $C$3-symbols: $C$4 and the unnormalized matrix is

$C$5

The transform is the Racah transform between two coupling schemes in $C$6 (Kubischta et al., 17 Apr 2026).

For permutation-invariant qudit codes, the MacWilliams matrix is an explicit finite Racah transform: $C$7 Special cases are

$C$8

The matrix satisfies

$C$9

Its rows are Racah orthogonal polynomials on the quadratic lattice

$\mathcal{L}(V)$0

and the eigenvalues of the degree-one twirl lie on the affine image

$\mathcal{L}(V)$1

with

$\mathcal{L}(V)$2

The tridiagonal multiplication rule

$\mathcal{L}(V)$3

yields a stable three-term recurrence for constructing transform rows (Teixeira, 14 May 2026).

4. Linear-programming formulation and truncation

In the multiplicity-free intrinsic setting, let

$\mathcal{L}(V)$4

for a code projector $\mathcal{L}(V)$5, and let $\mathcal{L}(V)$6, $\mathcal{L}(V)$7. The full feasibility system consists of

$\mathcal{L}(V)$8

together with detection equalities

$\mathcal{L}(V)$9

for the chosen detected sector set $$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$0 (Kubischta et al., 17 Apr 2026).

In normalized variables

$$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$1

one has

$$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$2

with positivity $$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$3, dominance $$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$4, and detection equalities $$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$5 on detected sectors.

Truncation retains only a subset $$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$6 of sectors. In the intrinsic quantum formulation, one introduces slack variables

$$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$7

subject to

$$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$8

and the truncated normalizations

$$g \cdot X := U(g) X U(g)^{-1}, \qquad X \in \mathcal{L}(V).$$9

For retained sectors,

$$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$0

The omitted contribution can be bounded linearly using nonnegativity and bounds on $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$1. This relaxation preserves validity of upper bounds because the feasible region enlarges, although tightness may be lost (Kubischta et al., 17 Apr 2026).

For the explicit truncated $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$2 system with $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$3 and indices $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$4, the retained variables are $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$5 and slacks $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$6, with constraints

$$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$7

$$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$8

where $$\mathcal{L}(V)=\bigoplus_{\xi \in \widehat G} W_\xi,$$9 absorbs omitted terms and may be bounded as

$\mathrm{SU}(q)$00

if row-wise bounds are known. Detection equalities are imposed for $\mathrm{SU}(q)$01, and choosing $\mathrm{SU}(q)$02 close to $\mathrm{SU}(q)$03 is reported to yield good bounds with small LPs (Kubischta et al., 17 Apr 2026).

The permutation-invariant qudit version uses variables $\mathrm{SU}(q)$04 with $\mathrm{SU}(q)$05 and dual variables

$\mathrm{SU}(q)$06

A truncated LP enforces only the constraints for $\mathrm{SU}(q)$07. 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 (Teixeira, 14 May 2026).

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 (Kubischta et al., 17 Apr 2026).

Example	Parameters	Unique feasible solution and certification
Four-qubit	$\mathrm{SU}(q)$08, $\mathrm{SU}(q)$09, $\mathrm{SU}(q)$10, $\mathrm{SU}(q)$11, detection set $\mathrm{SU}(q)$12	$\mathrm{SU}(q)$13, $\mathrm{SU}(q)$14; certifies extremality of the permutation-invariant code $\mathrm{SU}(q)$15
Seven-qubit	$\mathrm{SU}(q)$16, $\mathrm{SU}(q)$17, $\mathrm{SU}(q)$18, $\mathrm{SU}(q)$19, detection set $\mathrm{SU}(q)$20	$\mathrm{SU}(q)$21, $\mathrm{SU}(q)$22; certifies extremality of $\mathrm{SU}(q)$23
Three-qutrit	$\mathrm{SU}(q)$24, $\mathrm{SU}(q)$25, $\mathrm{SU}(q)$26, $\mathrm{SU}(q)$27, detection set $\mathrm{SU}(q)$28	$\mathrm{SU}(q)$29, $\mathrm{SU}(q)$30; certifies extremality of the permutation-invariant code $\mathrm{SU}(q)$31

In the four-qubit case, the constraints are

$\mathrm{SU}(q)$32

with

$\mathrm{SU}(q)$33

The unique feasible solution certifies that there is no $\mathrm{SU}(q)$34 for $\mathrm{SU}(q)$35, no $\mathrm{SU}(q)$36 with $\mathrm{SU}(q)$37, and no $\mathrm{SU}(q)$38 with $\mathrm{SU}(q)$39 (Kubischta et al., 17 Apr 2026).

In the seven-qubit case, the normalized system is

$\mathrm{SU}(q)$40

with $\mathrm{SU}(q)$41, $\mathrm{SU}(q)$42,

$\mathrm{SU}(q)$43

The unique feasible solution yields minimality of $\mathrm{SU}(q)$44 for $\mathrm{SU}(q)$45 with $\mathrm{SU}(q)$46, and excludes both $\mathrm{SU}(q)$47 with $\mathrm{SU}(q)$48 and $\mathrm{SU}(q)$49 with $\mathrm{SU}(q)$50 (Kubischta et al., 17 Apr 2026).

The qudit paper also gives a worked truncated example for $\mathrm{SU}(q)$51, $\mathrm{SU}(q)$52, with $\mathrm{SU}(q)$53, sector dimensions $\mathrm{SU}(q)$54, and truncation level $\mathrm{SU}(q)$55. The retained constraints are

$\mathrm{SU}(q)$56

$\mathrm{SU}(q)$57

$\mathrm{SU}(q)$58

with an example detection condition $\mathrm{SU}(q)$59 (Teixeira, 14 May 2026).

6. Multiplicity, semidefinite extension, and broader variants

When multiplicities occur, the intrinsic MacWilliams system ceases to be scalar. One fixes a decomposition

$\mathrm{SU}(q)$60

together with coherent isometric intertwiners $\mathrm{SU}(q)$61. The corresponding projector matrix units $\mathrm{SU}(q)$62 and twirl matrix units $\mathrm{SU}(q)$63 form orthogonal bases of the intertwiner algebra. The MacWilliams transform becomes block unitary, and the enumerators become matrix-valued: $\mathrm{SU}(q)$64

$\mathrm{SU}(q)$65

These matrices are Hermitian PSD blockwise and satisfy

$\mathrm{SU}(q)$66

If $\mathrm{SU}(q)$67 and $\mathrm{SU}(q)$68, then

$\mathrm{SU}(q)$69

and for a code projector $\mathrm{SU}(q)$70 of rank $\mathrm{SU}(q)$71,

$\mathrm{SU}(q)$72

The block MacWilliams identity is

$\mathrm{SU}(q)$73

so the feasibility problem is an SDP rather than an LP (Kubischta et al., 17 Apr 2026).

The intrinsic SDP constraints are

$\mathrm{SU}(q)$74

together with

$\mathrm{SU}(q)$75

In the first non-multiplicity-free $\mathrm{SU}(q)$76 example, with $\mathrm{SU}(q)$77 of dimension $\mathrm{SU}(q)$78, the decomposition of $\mathrm{SU}(q)$79 includes multiplicities such as $\mathrm{SU}(q)$80, $\mathrm{SU}(q)$81, $\mathrm{SU}(q)$82, $\mathrm{SU}(q)$83, and $\mathrm{SU}(q)$84. For the $\mathrm{SU}(q)$85-dimensional intrinsic code $\mathrm{SU}(q)$86 under restriction to the subgroup $\mathrm{SU}(q)$87, one obtains an SDP upper bound $\mathrm{SU}(q)$88 among depth-$\mathrm{SU}(q)$89 intrinsic codes, and the code attains the bound (Kubischta et al., 17 Apr 2026).

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 (Bariffi et al., 2024).

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 (Kubischta et al., 17 Apr 2026).