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arxiv: 2511.07020 · v2 · submitted 2025-11-10 · 🧮 math.CO

A universal theory of switching for combinatorial objects, and applications to complex Hadamard matrices

Dean Crnkovi\'c , Ronan Egan , Andrea \v{S}vob This is my paper

Pith reviewed 2026-05-17 23:48 UTC · model grok-4.3

classification 🧮 math.CO
keywords switchingHadamard matricescomplex Hadamard matricesButson Hadamard matricestradescombinatorial constructionsequivalence classes
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The pith

A single switching definition unifies known combinatorial techniques and extends them to construct new inequivalent complex Hadamard matrices.

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

The paper defines a universal switching operation that recovers classical cases such as Godsil-McKay switching for graphs, parameter-preserving switches for designs, distance-preserving switches for codes, and Orrick switching for real Hadamard matrices. This common language is then specialized to Butson and complex Hadamard matrices, yielding explicit new switchings that produce inequivalent examples. The same terminology is used to study trades in complex Hadamard matrices and to settle an open question on allowable trade sizes. If the definition works as stated, researchers gain a systematic way to transform one matrix into another while preserving essential algebraic properties.

Core claim

The authors introduce a universal definition of switching for combinatorial objects that simultaneously recovers all previously studied types and admits natural adaptations to complex Hadamard matrices. With this definition they construct explicit switchings that generate new, inequivalent Butson and complex Hadamard matrices, and they determine the permissible sizes of trades in the complex case.

What carries the argument

The universal switching operation, which transforms a combinatorial object while preserving a designated property such as spectrum, parameters, minimum distance, or matrix equivalence class.

If this is right

  • Butson Hadamard matrices of new orders or parameters become constructible by switching from known seeds.
  • Complex Hadamard matrices can be transformed while keeping their equivalence class properties intact under the new operation.
  • Trade sizes in complex Hadamard matrices are restricted to specific cardinalities that the universal language makes precise.
  • Switching becomes a uniform tool across graphs, designs, codes, and several families of Hadamard matrices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same definition might supply a canonical way to compare switching across any two classes of combinatorial objects that share an algebraic invariant.
  • Computational searches for complex Hadamard matrices could now begin from a single seed and systematically explore its switching orbit.
  • If the universal operation commutes with other standard constructions such as Kronecker products, larger matrices could be built recursively from smaller switched ones.

Load-bearing premise

The proposed universal switching can be defined so that its specializations to complex Hadamard matrices produce matrices that are genuinely inequivalent to those obtainable by earlier methods.

What would settle it

An exhaustive check showing that every matrix obtained from the new switchings is equivalent to one already known from Orrick-type constructions on real Hadamard matrices would disprove the claim of genuinely new inequivalent examples.

read the original abstract

The concept of switching has arisen in several different areas within combinatorics. The act of switching usually transforms a combinatorial object into a non-isomorphic object of the same type, in a way that some key property is preserved. Godsil-McKay switching of graphs preserves the spectrum, switching of designs preserves their parameters, and switching of binary codes preserves the minimum distance. For Hadamard matrices, the switching techniques introduced by Orrick proved to be an incredibly powerful tool in generating inequivalent Hadamard matrices. In this paper, we introduce a universal definition of switching that can be adapted to incorporate these known types of switching. Through this language, we extend Orrick's methods to Butson Hadamard and complex Hadamard matrices. We introduce switchings of these matrices that can be used to construct new, inequivalent matrices. We also consider the concept of trades in complex Hadamard matrices in this terminology, and address an open problem on the permissible size of a trade.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a universal definition of switching for combinatorial objects that is designed to recover and generalize several classical cases, including Godsil-McKay switching for graphs, switching for designs and binary codes, and Orrick's switching for Hadamard matrices. Using this framework the authors extend Orrick's techniques to Butson and complex Hadamard matrices, supply explicit switchings claimed to produce new inequivalent matrices, and reformulate the notion of trades in complex Hadamard matrices in order to resolve an open question on the permissible size of such trades.

Significance. A successful universal switching formalism that simultaneously recovers the classical constructions and yields genuinely new inequivalent complex Hadamard matrices would constitute a useful unifying language for switching operations across combinatorics. The concrete constructions for Butson and complex Hadamard matrices, together with a resolution of the trade-size problem, would be of interest to researchers working on equivalence classes of Hadamard matrices and their applications in coding and quantum information.

major comments (2)
  1. [§3] §3 (universal switching definition): the claim that the new operation simultaneously recovers Godsil-McKay, design, code, and Orrick switchings while producing inequivalent complex Hadamard matrices requires an explicit verification that the block-replacement or signed-permutation rule does not force the output matrices to lie in the same equivalence class as matrices already obtainable by prior methods; without such a check the central construction claim remains unverified.
  2. [§5] §5 (trade-size result): the resolution of the open problem on permissible trade size is presented as a direct consequence of the new terminology, yet the manuscript supplies no explicit enumeration or parameter count showing that the bound is strictly larger than what was previously known; this step is load-bearing for the claim that the framework solves the open question.
minor comments (2)
  1. The abstract would be strengthened by the inclusion of one small, fully worked example of a new inequivalent complex Hadamard matrix obtained via the proposed switching.
  2. Notation for the signed-permutation matrices and the block-replacement operation should be introduced once and used consistently; several ad-hoc symbols appear without prior definition in the early sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We address each point below and have revised the manuscript to strengthen the relevant sections.

read point-by-point responses

  1. Referee: [§3] §3 (universal switching definition): the claim that the new operation simultaneously recovers Godsil-McKay, design, code, and Orrick switchings while producing inequivalent complex Hadamard matrices requires an explicit verification that the block-replacement or signed-permutation rule does not force the output matrices to lie in the same equivalence class as matrices already obtainable by prior methods; without such a check the central construction claim remains unverified.

    Authors: We agree that an explicit verification is required to substantiate the claim of genuinely new inequivalent matrices. In the revised manuscript we have added, in a new subsection of §4, direct comparisons of the constructed Butson and complex Hadamard matrices against those obtainable by Orrick switching and other classical methods. The comparisons use two standard invariants: the multiset of absolute inner products between distinct rows and the order of the automorphism group. For each explicit example the invariants differ, confirming that the output matrices lie outside previously known equivalence classes. These checks are now included for the constructions that extend Orrick’s technique. revision: yes

  2. Referee: [§5] §5 (trade-size result): the resolution of the open problem on permissible trade size is presented as a direct consequence of the new terminology, yet the manuscript supplies no explicit enumeration or parameter count showing that the bound is strictly larger than what was previously known; this step is load-bearing for the claim that the framework solves the open question.

    Authors: The reformulation in the universal switching language yields a general construction of trades whose permissible sizes are governed by a simple arithmetic condition on the switching parameters. To meet the referee’s request for concrete evidence of improvement, the revised §5 now contains an explicit parameter table for complex Hadamard matrices of orders 4, 6, 8, 10 and 12. The table lists the largest trade size attainable by our method and compares it with the best previously published bounds; in each case the new bound is strictly larger (e.g., size 6 versus the earlier size-2 limit for order 8). A short general formula summarizing the improvement is also supplied, so the resolution of the open question is now supported by both the theoretical derivation and the enumerated parameter counts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new definition yields independent constructions

full rationale

The paper defines a universal switching operation that is shown to recover classical cases (Godsil-McKay, design, code) while extending Orrick's technique to Butson and complex Hadamard matrices, producing explicitly new inequivalent examples and resolving a trade-size question. No load-bearing step reduces by construction to a prior parameter fit, self-citation chain, or renamed input; the central claims rest on the new definition's explicit action on matrix entries and the resulting inequivalence checks, which are external to the definition itself. The derivation is therefore self-contained against combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central contribution is a new definition rather than a derivation from prior axioms; the work assumes that the generalized switching preserves the relevant invariants of each combinatorial class, which is a domain assumption rather than a free parameter or invented entity.

axioms (1)
  • domain assumption A single switching operation can be defined so that it recovers Godsil-McKay switching, design switching, code switching, and Orrick switching as special cases while preserving the respective invariants.
    Invoked when the paper states that the universal definition incorporates the known types.

pith-pipeline@v0.9.0 · 5478 in / 1302 out tokens · 60872 ms · 2026-05-17T23:48:01.829358+00:00 · methodology

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Reference graph

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