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On Constructing and Decoding Quantum Triorthogonal Codes

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read An integer linear program constructs even-weight triorthogonal matrices whose duals meet a target minimum distance by enforcing both overlap rules and Krawtchouk-polynomial conditions from the MacWilliams identities.

desk verdict New ILP for triorthogonal codes works in practice for the examples they found, but the loose link between weight enumerator and actual matrix rows needs explicit checks. read the letter →

arxiv 2605.24519 v1 pith:E2Q66YKQ submitted 2026-05-23 quant-ph cs.ITmath.IT

Alessio Baldelli , Olai {AA}. Mostad , Hsuan-Yin Lin , Eirik Rosnes , Massimo Battaglioni This is my paper · ORCID
classification quant-phcs.ITmath.IT
keywords triorthogonalcodesquantumCSSmagic-statedistillationintegerlinearprogrammingMacWilliamsidentitiesKrawtchoukpolynomialsdephasingchanneldecoding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an existence criterion that turns the search for even-weight triorthogonal generator matrices into an integer linear programming problem. The formulation joins the linear constraints that enforce pairwise and triple-wise row overlaps with additional linear conditions on the dual weight distribution, derived from MacWilliams identities via Krawtchouk polynomials. If a feasible solution exists, the resulting matrix is guaranteed to be triorthogonal and to produce a dual code whose minimum distance meets the prescribed target. The same framework yields previously unknown triorthogonal codes that need not arise from classical triply-even codes. The work also evaluates decoding of high-distance codes obtained by the doubling construction over the dephasing channel, comparing bounded-distance, belief-propagation, and GRAND-based decoders.

What carries the argument

Integer linear programming formulation that augments the triorthogonality linear constraints with Krawtchouk-polynomial conditions on the dual weight distribution obtained from the MacWilliams identities.

What would settle it

An explicit matrix produced by the integer program that satisfies all listed linear constraints yet whose dual code has minimum distance strictly smaller than the target value, or that fails to be triorthogonal.

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Extended reading notes

Core claim

Triorthogonality constraints together with Krawtchouk-polynomial conditions on the dual weight distribution, when solved as an integer linear program, produce even-weight triorthogonal generator matrices whose duals achieve any chosen minimum distance; the method yields new nontrivial codes and enables concrete decoding comparisons for doubled high-distance instances over the dephasing channel.

Load-bearing premise

The Krawtchouk-polynomial conditions derived from the MacWilliams identities, when added to the triorthogonality constraints, are enough to ensure that every feasible integer solution corresponds to a valid triorthogonal matrix whose dual has the target minimum distance.

Editorial extensions

If this is right

  • New triorthogonal codes exist that are not generated by classical triply-even codes.
  • The doubling construction produces high-distance triorthogonal codes whose decoding performance over the dephasing channel can be compared across bounded-distance, belief-propagation, and GRAND decoders.
  • The GRAND-based decoder adapted to the quantum setting emerges as a competitive option for these codes.
  • The integer-linear-programming existence criterion can be applied to search for even-weight triorthogonal matrices at larger block lengths.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same linear-programming approach could be tested on other CSS-code families that require simultaneous linear constraints and weight-distribution targets.
  • If the GRAND decoder remains strong, it could be combined with the new triorthogonal matrices to improve magic-state distillation thresholds.
  • The existence criterion might be relaxed or tightened by replacing the Krawtchouk conditions with direct minimum-distance constraints when the block length remains modest.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper derives an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance by combining triorthogonality constraints with MacWilliams identities expressed via Krawtchouk-polynomial conditions on the dual weight distribution; this yields an integer linear programming (ILP) formulation. The work reports discovery of new nontrivial triorthogonal codes not necessarily arising from classical triply-even codes and evaluates decoding performance (bounded-distance, BP+OSD, and GRAND) of high-distance codes from the doubling construction over the dephasing channel.

Significance. If the ILP solutions are guaranteed to produce valid matrices, the criterion supplies a systematic, non-ad-hoc method for constructing triorthogonal CSS codes with controlled dual distance, extending beyond the classical triply-even route and thereby supporting magic-state distillation. The decoding comparisons supply concrete performance data for these codes. The combination of linear-programming techniques with quantum coding constraints is a clear methodological contribution.

major comments (2)
  1. [existence criterion] Existence criterion section: the manuscript asserts that the triorthogonality linear constraints together with the Krawtchouk-polynomial MacWilliams conditions are jointly sufficient for every feasible ILP solution to correspond to a realizable even-weight triorthogonal matrix whose dual has the prescribed minimum distance. The weight-enumerator variables, however, remain only loosely coupled to the explicit row-overlap and row-weight variables; no argument is supplied showing that an integer solution for the enumerator is always realizable by a matrix satisfying the overlap equations. This sufficiency gap is load-bearing for the central claim.
  2. [abstract] Abstract and construction results: the claim of having found 'new nontrivial triorthogonal codes' is stated without exhibiting any explicit generator matrices, their dimensions, or verification that the obtained ILP solutions satisfy all row-weight and overlap conditions simultaneously. Concrete examples would directly test whether the ILP produces valid codes and would allow readers to assess the practical reach of the criterion.
minor comments (2)
  1. [abstract] The abstract lists three decoders but does not indicate which channel model parameters or code distances were used in the numerical comparisons; adding a brief statement of the simulation settings would improve readability.
  2. Notation for the dual weight distribution and the Krawtchouk polynomials could be introduced once in a dedicated preliminary subsection rather than inline, to aid readers unfamiliar with the MacWilliams route.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments identify important points regarding the rigor of the existence criterion and the presentation of concrete results. We address each major comment below and will revise the manuscript to strengthen these aspects.

read point-by-point responses

  1. Referee: [existence criterion] Existence criterion section: the manuscript asserts that the triorthogonality linear constraints together with the Krawtchouk-polynomial MacWilliams conditions are jointly sufficient for every feasible ILP solution to correspond to a realizable even-weight triorthogonal matrix whose dual has the prescribed minimum distance. The weight-enumerator variables, however, remain only loosely coupled to the explicit row-overlap and row-weight variables; no argument is supplied showing that an integer solution for the enumerator is always realizable by a matrix satisfying the overlap equations. This sufficiency gap is load-bearing for the central claim.

    Authors: We thank the referee for identifying this gap. The ILP formulation incorporates the triorthogonality constraints on row weights and pairwise/triple overlaps together with the MacWilliams identities expressed via Krawtchouk polynomials on the dual weight distribution. While feasible integer solutions are expected to yield valid matrices, the manuscript does not supply an explicit argument proving that every solution to the enumerator variables is realizable by a matrix satisfying the overlap equations. In the revised version we will add a dedicated subsection establishing this sufficiency, either by a direct realizability proof or by exhibiting explicit matrix constructions for all ILP solutions reported. This will make the central claim fully rigorous. revision: yes

  2. Referee: [abstract] Abstract and construction results: the claim of having found 'new nontrivial triorthogonal codes' is stated without exhibiting any explicit generator matrices, their dimensions, or verification that the obtained ILP solutions satisfy all row-weight and overlap conditions simultaneously. Concrete examples would directly test whether the ILP produces valid codes and would allow readers to assess the practical reach of the criterion.

    Authors: We agree that explicit examples are necessary to substantiate the claim of new nontrivial triorthogonal codes. Although the abstract and results section report the discovery of such codes via the ILP, the submitted manuscript does not display the generator matrices, their dimensions, or direct verification of the overlap conditions. In the revision we will include concrete generator matrices for the new codes, together with their parameters and explicit checks confirming that all triorthogonality and dual-distance conditions are satisfied. These examples will allow readers to verify the ILP outputs and evaluate the method's reach. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation combines external identities with linear constraints

full rationale

The paper's existence criterion is obtained by adjoining the standard triorthogonality linear equations on row overlaps to the MacWilliams identities expressed via Krawtchouk polynomials on the dual weight distribution. Both sets of relations are drawn from classical coding theory and are independent of the present work; the resulting ILP is a search procedure whose feasible points are asserted to realize the desired matrices. No step equates a derived quantity to a fitted parameter, renames an input as a prediction, or relies on a load-bearing self-citation whose validity is presupposed by the authors. The derivation chain therefore remains self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the standard MacWilliams identities (external) and the assumption that the Krawtchouk conditions plus triorthogonality linear constraints are jointly sufficient; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

assumptions (2)
  • standard math MacWilliams identities hold for the dual weight distribution of binary codes
    Invoked to translate dual-distance target into Krawtchouk-polynomial conditions on the weight enumerator.
  • domain assumption Triorthogonality is expressible as a set of linear constraints over GF(2)
    Used to formulate the generator-matrix search as an ILP.

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Cite this review

Pith. "Pith review of On Constructing and Decoding Quantum Triorthogonal Codes." pith.science (2026). https://pith.science/paper/E2Q66YKQ

@misc{pith2026260524519,
  author       = {Pith},
  title        = {Pith review of: On Constructing and Decoding Quantum Triorthogonal Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E2Q66YKQ}},
  note         = {Machine review of arXiv:2605.24519}
}
abstract

A triorthogonal code is a binary quantum Calderbank-Shor-Steane (CSS) code defined by a triorthogonal matrix. Triorthogonal codes are a key ingredient in magic-state distillation, since they allow for transversal $\mathsf{T}$ gates, a non-Clifford logical operation useful for achieving universal fault-tolerant quantum computation. Their construction is challenging because it must satisfy simultaneous pairwise and triple-wise overlap constraints, as well as row-weight requirements. In this work, we study the construction and decoding of triorthogonal codes with prescribed dual-distance properties. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find new nontrivial triorthogonal codes that are not necessarily generated by classical triply-even codes. The decoding performance of high-distance triorthogonal codes obtained via the doubling construction is then evaluated over the dephasing channel. We compare bounded-distance decoding, belief propagation plus ordered-statistics post-processing, and a GRAND-based decoder adapted to the quantum setting, which turns out to be a promising option.

Figures

Figures reproduced from arXiv: 2605.24519 by the authors.

Figure 1
Figure 1. Subfigs. (a) and (b): LER of two triorthogonal codes on the dephasing channel for several decoders. Subfig. (c): Comparison of the estimated average [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exploring the landscape of compact magic-state distillation factories

    quant-ph 2026-06 unverdicted novelty 8.0 of 10

    Classical repetition-code framing plus SAT search yields no-go theorems (no d>3 T-to-T on <8 qubits) and the smallest known unitary factories for d=4,5 T-states (10–11 qubits) and d=3,4 CCZ-states (9–10 qubits).

Reference graph

Works this paper leans on

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