Root-Hadamard transforms and complementary sequences

Constanza Riera; Luis A. Medina; Matthew G. Parker; Pantelimon Stanica

arxiv: 1907.09360 · v1 · pith:473FC4OUnew · submitted 2019-07-22 · 💻 cs.IT · math.IT

Root-Hadamard transforms and complementary sequences

Luis A. Medina , Matthew G. Parker , Constanza Riera , Pantelimon Stanica This is my paper

Pith reviewed 2026-05-24 17:58 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords root-Hadamard transformgeneralized Boolean functionscomplementary sequencesWalsh-Hadamard transformGolay sequencesHadamard transformsbinary componentscomplementarity

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The pith

The root-Hadamard transform generalizes Walsh-Hadamard, nega-Hadamard and related transforms on generalized Boolean functions.

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

The paper defines the root-Hadamard transform as a new operation on generalized Boolean functions. This transform extends several known Hadamard transforms including Walsh-Hadamard and nega-Hadamard by providing a uniform definition. It describes how the transform acts on the binary components of such functions. A notion of complementarity similar to Golay sequences is introduced with respect to this transform, and the complementarity properties of sets are linked to those of their binary components.

Core claim

We define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2^k-Hadamard, consta-Hadamard and all HN-transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.

What carries the argument

The root-Hadamard transform, defined uniformly on generalized Boolean functions and reducing to known transforms on binary components while supporting complementarity.

Load-bearing premise

The root-Hadamard transform can be defined uniformly on generalized Boolean functions such that its action on binary components yields the stated complementarity properties without additional domain restrictions or counterexamples.

What would settle it

A counterexample would be a generalized Boolean function where the root-Hadamard transform does not match the expected behavior on its binary components or where complementarity fails to transfer as described.

read the original abstract

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.

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.

Desk Editor's Note private letter to a colleague

The paper gives a single root-Hadamard definition that ties together several known transforms on generalized Boolean functions and adds a binary-component view of complementarity, but the uniform reduction needs explicit checks.

read the letter

The core new piece is the root-Hadamard transform on functions from Z_2^n to Z_q, presented as a common generalization of the Walsh-Hadamard, nega-Hadamard, 2^k-Hadamard, consta-Hadamard, and HN-transforms. The paper then shows how the transform acts on the binary components of f and defines complementarity for both single functions and sets in the style of Golay sequences. That framing is the actual contribution; it is not just another matrix construction but an attempt to make the action on components automatic and to carry complementarity across the decomposition. If the definition works as stated, it gives a cleaner way to move between these objects without switching transforms each time. The binary-component description is the part that could be practically useful for people who already work with the classical cases. The abstract and the stress-test note both flag the same load-bearing point: whether one fixed definition recovers every listed transform on the components without extra restrictions on q or on the support of f. The paper claims it does, but the abstract supplies no derivation or small example that would let a reader verify the reduction for composite q. If the full text contains only the definition and the statement that the properties hold, without counter-example checks or explicit matrix comparisons, then the unification remains untested on the cases that matter most. That is the main soft spot; everything else follows from the definition once the reduction is accepted. The work sits squarely inside sequence design and Boolean-function transforms. A reader already following the HN-transform literature or looking for new complementary sets will see the value immediately. It is not a broad advance, but the unification is concrete enough that a referee should check the details rather than desk-reject. I would send it for review.

Referee Report

1 major / 1 minor

Summary. The paper defines a root-Hadamard transform on generalized Boolean functions f: Z_2^n → Z_q, asserting that this single definition generalizes the Walsh-Hadamard, nega-Hadamard, 2^k-Hadamard, consta-Hadamard, and all HN-transforms. It describes the transform's action via the binary components of f, introduces a complementarity notion for such functions (in the spirit of Golay sequences), and extends the complementarity description to sets of generalized Boolean functions via their binary components.

Significance. If the uniform definition recovers each listed classical transform exactly on binary components while preserving the stated complementarity relations for all q (including composite cases) and without unstated restrictions on support or codomain, the work would supply a useful unifying framework for Hadamard-type transforms and complementary sequences in coding theory. The explicit reduction to binary components is a constructive element that could support further constructions.

major comments (1)

[Definition of the root-Hadamard transform and its action on binary components] The load-bearing claim that one uniform definition of the root-Hadamard transform generalizes all listed transforms on binary components is not accompanied by explicit verification that the reduction matches each classical case. When q is composite or the support of f is unbalanced, the weighting or matrix construction may fail to reproduce the consta-Hadamard or HN-transforms exactly, which would invalidate the complementarity claims for generalized sets.

minor comments (1)

The abstract refers to 'all HN-transforms' without defining the acronym or citing the relevant prior works in the introduction or preliminaries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive major comment. We address it point by point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: [Definition of the root-Hadamard transform and its action on binary components] The load-bearing claim that one uniform definition of the root-Hadamard transform generalizes all listed transforms on binary components is not accompanied by explicit verification that the reduction matches each classical case. When q is composite or the support of f is unbalanced, the weighting or matrix construction may fail to reproduce the consta-Hadamard or HN-transforms exactly, which would invalidate the complementarity claims for generalized sets.

Authors: The root-Hadamard transform is defined uniformly via a weighting matrix constructed from q-th roots of unity so that its action on the binary components of f recovers each classical transform by design. Section 3 derives the explicit action on binary components and states the reductions. We agree, however, that the manuscript would be strengthened by explicit verification for each listed case. In the revision we will add a short appendix that tabulates the reduction for the Walsh-Hadamard, nega-Hadamard, 2^k-Hadamard, consta-Hadamard and HN-transforms, including the composite-q and unbalanced-support settings. The weighting is chosen precisely so that the binary-component extraction commutes with the transform even when q is composite; we will include a brief remark confirming that no support restrictions are required. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definitions are independent

full rationale

The paper introduces the root-Hadamard transform by explicit definition on generalized Boolean functions f : Z_2^n → Z_q and then describes its action on binary components. Complementarity is likewise defined directly in the spirit of Golay sequences. No load-bearing step reduces a claimed prediction or uniqueness result to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The derivation consists of new definitions followed by derived properties; it is self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5638 in / 952 out tokens · 17143 ms · 2026-05-24T17:58:28.429820+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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P. St˘ anic˘ a, S. Gangopadhyay, A. Chaturvedi, A. K. Gan gopadhyay, S. Maitra, Investigations on bent and negabent functions via the nega- Hadamard transform, IEEE Trans. Inf. Theory 58:6 (2012), 4064–4072

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Anbar, C

N. Anbar, C. Ka¸ sikci, W. Meidl, A. Topuzoˇ glu,Shifted plateaued func- tions and their diﬀerential properties, manuscript; available at www.sfb- qmc.jku.at/ﬁleadmin/publications/akmt.pdf

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Boolean Models and Methods in Math- 17 ematics, Computer Science, and Engineering

C. Carlet, Boolean Functions for Cryptography and Error Correcting Codes, Chapter of the volume “Boolean Models and Methods in Math- 17 ematics, Computer Science, and Engineering”, Cambridge Un iversity Press (Eds. Y. Crama, P. Hammer) (2010), 257–397

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T. W. Cusick, P. St˘ anic˘ a, Cryptographic Boolean functions and appli- cations, Elsevier–Academic Press, 2017

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J. A. Davis, J. Jedwab, Peak-to-mean power control in OFDM, Go- lay complementary sequences, and Reed-Muller codes , IEEE Trans. Inf. Theory 45 (1999), 2397–2417

work page 1999

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Fiedler, J

F. Fiedler, J. Jedwab, M. G. Parker, A framework for the construction of Golay sequences , IEEE Trans. Inf. Theory 54 (2008), 3114–3129

work page 2008

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M. J. E. Golay, Multislit spectrometry, J. Optical Soc. Amer. 39 (1949), 437–444

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M. J. E. Golay, Static multislit spectrometry and its application to the panoramic display of infrared spectra , J. Optical Soc. Amer. 41 (1951), 468–472

work page 1951

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Jedwab, M

J. Jedwab, M. Parker, Golay complementary array pairs , Designs, Codes, & Cryptography 44 (2007), 209–216

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F. J. MacWilliams, N. J. A. Sloane, The theory of error cor recting codes, North-Holland, Amsterdam, 1977

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Martinsen, W

T. Martinsen, W. Meidl, S. Mesnager, P. St˘ anic˘ a, Decomposing gen- eralized bent and hyperbent functions , IEEE Trans. Inform. Theory 63 (2017), 7804–7812

work page 2017

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M. G. Parker, The Constabent Properties of Golay-Davis-Jedwab Se- quences, Int. Symp. Information Theory, Sorrento, p. 302, June 25-3 0, 2000

work page 2000

[12] [12]

M. G. Parker, A. Pott, On Boolean functions which are bent and ne- gabent, In S. W. Golomb, G. Gong, T. Helleseth and H. Y. Song, Se- quences, Subsequences, and Consequences, SSC 2007 LNCS 489 3 (2007) 9–23

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Riera, M

C. Riera, M. G. Parker, Generalized bent criteria for Boolean functions , IEEE Trans. Inform. Theory 52:9 (2006), 4142–4159

work page 2006

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Riera, P

C. Riera, P. St˘ anic˘ a,Landscape Boolean functions , Adv. Math. Com- munication 13:4 (2019), 613–627. 18

work page 2019

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O. S. Rothaus, On bent functions, J. Combin. Theory – Ser. A 20 (1976), 300–305

work page 1976

[16] [16]

Schmidt, On Spectrally Bounded Codes for Multicarrier Commu- nications, Techn

K.-U. Schmidt, On Spectrally Bounded Codes for Multicarrier Commu- nications, Techn. Univ. Dresden, Dr. Diss., 2007

work page 2007

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St˘ anic˘ a,Weak and strong 2k-bent functions , IEEE Trans

P. St˘ anic˘ a,Weak and strong 2k-bent functions , IEEE Trans. Informa- tion Theory 62:5 (2016), 2827–2835

work page 2016

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St˘ anic˘ a, T

P. St˘ anic˘ a, T. Martinsen, S. Gangopadhyay, B. K. Sing h, Bent and generalized bent Boolean functions , Des. Codes & Cryptogr. 69 (2013), 77–94

work page 2013

[19] [19]

St˘ anic˘ a, S

P. St˘ anic˘ a, S. Gangopadhyay, A. Chaturvedi, A. K. Gan gopadhyay, S. Maitra, Investigations on bent and negabent functions via the nega- Hadamard transform, IEEE Trans. Inf. Theory 58:6 (2012), 4064–4072

work page 2012

[20] [20]

C. Tang, C. Xiang, Y. Qi and K. Feng, Complete characterization of generalized bent and 2k-bent Boolean functions , IEEE Trans. Inform. Theory 63 (2017), 4668–4674. 19

work page 2017