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arxiv: 2510.12315 · v2 · submitted 2025-10-14 · 📡 eess.SP

Systematic Constructions of Complementary Sets and Hadamard Matrices from Circulant Operator

Piyush Priyanshu , Sudhan Majhi , Subhabrata Paul This is my paper

Pith reviewed 2026-05-18 08:09 UTC · model grok-4.3

classification 📡 eess.SP
keywords Hadamard matricesGolay complementary setscross Z-complementary setscomplete complementary codescirculant operatorszero correlation zonesequence construction
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The pith

Arbitrary Hadamard matrices act as seeds to generate Golay complementary sets and optimal cross-Z complementary sequence sets through circulant algebraic transformations.

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

The paper develops a systematic algebraic framework based on circulant operators that starts from Hadamard matrices of any order to produce several families of complementary sequences. These include Golay complementary sets, cross Z-complementary sets with controlled zero-correlation zones, complete complementary codes, and optimal cross-Z complementary sequence sets. A reader would find value in the ability to construct these for arbitrary lengths and with flexible user capacities, which supports better designs in signal processing applications. The work also provides a theoretical connection between Hadamard matrices and Golay complementary pairs.

Core claim

This work establishes that arbitrary Hadamard matrices can be used as flexible seeds to systematically generate Golay complementary sets, cross Z-complementary sets, complete complementary codes, and optimal cross-Z complementary sequence sets through algebraic transformations using cyclic operators. Specifically, circulant Hadamard matrices of order 4 yield binary CZCS of arbitrary lengths with maximum ZCZ ratio of 2/3, while Hadamard matrices of any order give CZCS with ZCZ ratio 1/2. Further constructions produce CCC and optimal CZCSS, marking the first generalized framework for optimal CZCSS from arbitrary Hadamard seeds.

What carries the argument

Cyclic operators applied to Hadamard matrices that transform them into sequence sets while maintaining correlation zone properties.

If this is right

  • Binary CZCS of arbitrary lengths achieve a ZCZ ratio of 2/3 using recursive circulant Hadamard matrices of order 4.
  • Any order binary or complex Hadamard matrices produce CZCS with ZCZ ratio of 1/2.
  • Binary GCS of all lengths and Hadamard matrices of order 2^{a+1} 10^b 26^c are constructed using circulant matrices and Golay pairs.
  • Binary CCC with parameters (2N, 2N, 2N) where N=2^a 10^b 26^c and optimal binary (8n, 8n, 8n, 4n)-CZCSS are obtained.

Where Pith is reading between the lines

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

  • The Hadamard-to-GCS relation may support new constructions for other correlation-controlled objects in coding theory.
  • The arbitrary-length property could enable sequence designs for variable-rate multi-user systems without fixed block constraints.
  • Implementation tests in radar or CDMA scenarios would show whether the ZCZ ratios translate to measurable interference reduction.

Load-bearing premise

The algebraic transformations on the Hadamard matrices preserve the zero-correlation-zone properties at the stated ratios for sequences of arbitrary lengths.

What would settle it

A direct computation of the aperiodic correlation functions for a constructed CZCS of length 12 from a Hadamard matrix of order 6 to check whether the zero-correlation zone reaches exactly half the length without extra correlation peaks.

Figures

Figures reproduced from arXiv: 2510.12315 by Piyush Priyanshu, Subhabrata Paul, Sudhan Majhi.

Figure 1
Figure 1. Figure 1: This represents the relation among CZCS, GCS and Hada [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: This represents the relation between GCS and Hadamar [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sum of AACF of each row of Ri ⊙ G -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Magnitude of ACCF -20 -15 -10 -5 0 5 10 15 20 Time shift [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sum of ACCF between each rows of Ri ⊙ G and Rj ⊙ G [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

A Hadamard matrix $H$ of order $n$ is a square matrix with entries $\pm 1$ satisfying $HH^T = nI_n$, where $I_n$ is the identity matrix of order $n$. A circulant Hadamard matrix is a Hadamard matrix whose rows are cyclic shifts of one another. This work establishes a unified algebraic framework that treats arbitrary Hadamard matrices as flexible seeds to systematically generate Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS) through algebraic transformations. In this paper, a systematic framework using cyclic operators is presented. First, circulant Hadamard matrices of order 4 are utilized recursively to propose binary CZCS of arbitrary lengths, achieving a maximum ZCZ ratio of 2/3, and binary GCS. Significantly, this framework is generalized to establish that by employing binary or complex Hadamard matrices of any order, binary or non-binary CZCSs of arbitrary lengths can be constructed with a ZCZ ratio of 1/2. Furthermore, to provide flexible user capacity, an alternative construction of binary GCS of all lengths and Hadamard matrices of order $2^{a+1} 10^b 26^c$ ($a, b, c \geq 0$) is proposed using circulant matrices and Golay complementary pairs (GCP). These constructions are further extended to form binary CCC with parameters $(2N, 2N, 2N)$, where $N=2^a 10^b 26^c$, and $(4n, 4n, 4n)$ for $n \geq 1$. Additionally, optimal binary $(8n, 8n, 8n, 4n)$-CZCSS and their complex versions with parameters $(4m, 4m, 4m, 2m)$ are proposed for $n, m \geq 1$. These results provide the first generalized framework for constructing optimal CZCSS from arbitrary Hadamard seeds. Finally, a theoretical relation between Hadamard matrices and GCSs is established, and fundamental properties of circulant matrices over aperiodic correlation functions are presented.

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 / 1 minor

Summary. The manuscript presents a unified algebraic framework that employs circulant operators applied to arbitrary Hadamard matrices (binary or complex) as seeds to construct Golay complementary sets (GCS), cross Z-complementary sets (CZCS), complete complementary codes (CCC), and optimal cross-Z complementary sequence sets (CZCSS). Specific claims include recursive constructions of binary CZCS of arbitrary lengths with maximum ZCZ ratio 2/3 from circulant Hadamard matrices of order 4, generalized constructions achieving ZCZ ratio 1/2 from Hadamard matrices of any order, binary GCS and CCC with parameters such as (2N,2N,2N) for N=2^a 10^b 26^c and (4n,4n,4n), and optimal binary (8n,8n,8n,4n)-CZCSS together with complex (4m,4m,4m,2m) versions; a theoretical relation between Hadamard matrices and GCSs is also established.

Significance. If the algebraic transformations are shown to preserve the required zero sums of aperiodic cross-correlations inside the claimed ZCZ widths for arbitrary lengths, the work would constitute a significant contribution by supplying the first generalized framework for optimal CZCSS from arbitrary Hadamard seeds. The recursive and order-independent constructions, together with explicit parameter families for CCC and CZCSS, would offer practical flexibility for applications in CDMA, radar, and sequence design.

major comments (2)
  1. [Recursive construction of binary CZCS] The central claim that recursive application of the circulant operator to order-4 circulant Hadamard matrices produces binary CZCS with ZCZ ratio exactly 2/3 for every integer length (not merely multiples of 4) requires an explicit length-independent argument. The abstract states that the transformations achieve this ratio but supplies no derivation showing that the resulting aperiodic cross-correlation sums remain identically zero inside the zone once recursion depth is fixed; any length-dependent boundary terms would collapse the ratio claim.
  2. [Generalized framework for CZCS and CZCSS] In the generalization to arbitrary-order Hadamard seeds, the mapping from HH^T = nI to zero aperiodic cross-correlations over a ZCZ of width exactly half the sequence length is asserted for any length, yet no concrete verification or inductive step is indicated that rules out length-dependent residuals when the seed order does not divide the target length.
minor comments (1)
  1. [Construction parameters] The notation for CCC parameters (2N,2N,2N) and CZCSS (8n,8n,8n,4n) would benefit from an explicit statement of how N and n relate to the underlying sequence length and the number of sets.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below with clarifications drawn from the existing proofs and indicate the revisions we will make to improve explicitness and readability.

read point-by-point responses

  1. Referee: [Recursive construction of binary CZCS] The central claim that recursive application of the circulant operator to order-4 circulant Hadamard matrices produces binary CZCS with ZCZ ratio exactly 2/3 for every integer length (not merely multiples of 4) requires an explicit length-independent argument. The abstract states that the transformations achieve this ratio but supplies no derivation showing that the resulting aperiodic cross-correlation sums remain identically zero inside the zone once recursion depth is fixed; any length-dependent boundary terms would collapse the ratio claim.

    Authors: We thank the referee for this observation. The recursive construction and its length-independent ZCZ-ratio preservation are established in Section III via induction on recursion depth: the base circulant Hadamard matrix of order 4 supplies the zero-sum property for aperiodic cross-correlations inside the zone of width (2/3) of the current length, and each application of the circulant operator maps the preceding zero sums to new zero sums without introducing length-dependent boundary residuals, because the operator is defined uniformly via cyclic shifts that commute with the correlation summation. The induction step therefore holds for any integer length obtained after any finite number of recursions. To make this argument more self-contained and to preempt any concern about boundary terms, we will add an explicit lemma (new Lemma 3.2) that isolates the length-invariance claim and supplies the short inductive verification. revision: yes

  2. Referee: [Generalized framework for CZCS and CZCSS] In the generalization to arbitrary-order Hadamard seeds, the mapping from HH^T = nI to zero aperiodic cross-correlations over a ZCZ of width exactly half the sequence length is asserted for any length, yet no concrete verification or inductive step is indicated that rules out length-dependent residuals when the seed order does not divide the target length.

    Authors: The referee correctly notes that the generalized construction would benefit from an explicit verification when the seed order does not divide the target length. The mapping itself follows directly from the defining relation HH^T = nI together with the algebraic action of the circulant operator, which converts the row-orthogonality into the required zero sums of aperiodic cross-correlations over a zone of width exactly half the final length; this identity holds irrespective of divisibility because the operator is applied entry-wise after embedding the seed into a larger circulant structure. Nevertheless, we agree that a short inductive or direct verification for the non-dividing case would strengthen the presentation. We will therefore insert a new paragraph and a supporting corollary in Section IV that explicitly treats the non-dividing case and confirms the absence of residual terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions are explicit algebraic mappings from standard Hadamard and circulant properties.

full rationale

The paper defines circulant Hadamard matrices via the standard equation HH^T = nI and applies recursive cyclic operators to generate GCS, CZCS, CCC, and CZCSS. These steps are direct transformations whose correlation-zero properties follow from the seed matrix definition and the algebraic rules of the operator, without reducing any claimed prediction or ratio (2/3 or 1/2) to a fitted parameter or self-citation that presupposes the target result. No uniqueness theorem is imported from the authors' prior work, no ansatz is smuggled via citation, and no known empirical pattern is merely renamed. The framework is self-contained against external benchmarks of Hadamard orthogonality and aperiodic correlation, yielding a non-circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The constructions rest on standard algebraic properties of Hadamard matrices and circulant operators; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • domain assumption Hadamard matrices satisfy HH^T = n I_n and can be used as seeds for algebraic transformations that preserve correlation properties.
    Invoked when treating arbitrary Hadamard matrices as flexible seeds for generating GCS, CZCS, CCC, and CZCSS.
  • domain assumption Circulant operators applied to Hadamard matrices of order 4 or arbitrary order produce sequences with the stated ZCZ ratios for any length.
    Central to the recursive and generalized constructions of binary and complex CZCS.

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