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arxiv: 2606.29950 · v1 · pith:72VGI4E2new · submitted 2026-06-29 · 💻 cs.IT · math.IT

New families of asymptotically optimal codebooks from vectorial dual-bent functions

Yadi Wei , Jiaxin Wang , Fang-Wei Fu , Wenjuan Yin This is my paper

Pith reviewed 2026-06-30 04:19 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords codebooksWelch boundvectorial dual-bent functionscross-correlationasymptotically optimalCDMAbent functions
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The pith

Vectorial dual-bent functions yield new codebook families that asymptotically achieve the Welch bound.

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

The paper establishes constructions of codebooks from vectorial dual-bent functions. These constructions produce families where the maximum cross-correlation amplitudes approach the Welch bound for large parameters. Explicit formulas are given for the maximum amplitudes and the distributions of all cross-correlation values. The resulting codebooks include ones with previously unseen parameters and some with small alphabet sizes.

Core claim

By using vectorial dual-bent functions, several families of codebooks are constructed that asymptotically achieve the Welch bound. The maximum cross-correlation amplitudes and the distributions of the cross-correlation amplitudes of the constructed codebooks are explicitly determined. Furthermore, these codebooks have new parameters, and some of them have very small alphabet sizes.

What carries the argument

Vectorial dual-bent functions used to generate the codebook entries with controlled correlations.

If this is right

  • The codebooks are suitable for use in CDMA communication systems due to their correlation properties.
  • Explicit distributions support applications in compressed sensing.
  • New parameters provide additional options for code design in MIMO communications.
  • Small alphabet sizes may simplify hardware implementations in coding applications.

Where Pith is reading between the lines

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

  • Further families might be obtained by considering other types of vectorial functions beyond dual-bent ones.
  • The constructions could be evaluated numerically for moderate sizes to check how quickly they approach the bound.
  • Links to existing bent function literature may allow algebraic simplifications or generalizations.

Load-bearing premise

Such vectorial dual-bent functions with the properties needed for the correlation bounds exist and map appropriately to the codebook vectors.

What would settle it

Finding a vectorial dual-bent function that generates a codebook whose maximum cross-correlation amplitude fails to approach the Welch bound.

read the original abstract

Codebooks with small maximum cross-correlation amplitudes play an important role in many applications, such as code division multiple access (CDMA) communication systems, multiple-input multiple-output (MIMO) communications, compressed sensing, and coding theory. In this paper, by using vectorial dual-bent functions, we construct several families of codebooks that asymptotically achieve the Welch bound. The maximum cross-correlation amplitudes and the distributions of the cross-correlation amplitudes of the constructed codebooks are explicitly determined. Furthermore, these codebooks have new parameters, and some of them have very small alphabet sizes.

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 constructs several families of codebooks from vectorial dual-bent functions. These families are claimed to asymptotically achieve the Welch bound, with explicit determination of the maximum cross-correlation amplitudes and the full distributions of the cross-correlation amplitudes. The constructions are asserted to yield new parameters, some with very small alphabet sizes, for applications including CDMA and MIMO.

Significance. If the existence of the required vectorial dual-bent functions and the exact preservation of the correlation formulas under the mapping are established, the work would supply new asymptotically optimal codebook families with potentially advantageous parameters and alphabet sizes. The explicit correlation distributions would enable precise performance evaluation beyond the maximum amplitude alone.

major comments (2)
  1. [§3] §3 (Constructions): The central claim that the families asymptotically achieve the Welch bound rests on the existence of vectorial dual-bent functions with the precise parameters needed for the mapping to codebook vectors; the manuscript must supply either explicit constructions or a self-contained existence proof for these functions in the stated dimensions, as the abstract provides no verification steps.
  2. [§4] §4 (Correlation analysis): The derivation that the vectorial dual-bent property directly yields the stated maximum cross-correlation amplitude and amplitude distribution must be checked for any additional cross terms introduced by the vectorial embedding; if the reduction to the scalar bent case is not exact, the explicit formulas and asymptotic optimality do not hold.
minor comments (1)
  1. The abstract could include a brief parenthetical reference to the specific prior bent-function literature used for the vectorial dual constructions to aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments, which help clarify the presentation of our constructions and analysis. We address each major comment below and will revise the manuscript accordingly to include the requested explicit constructions and verification steps.

read point-by-point responses

  1. Referee: [§3] §3 (Constructions): The central claim that the families asymptotically achieve the Welch bound rests on the existence of vectorial dual-bent functions with the precise parameters needed for the mapping to codebook vectors; the manuscript must supply either explicit constructions or a self-contained existence proof for these functions in the stated dimensions, as the abstract provides no verification steps.

    Authors: We agree that explicit verification strengthens the central claim. In the revised manuscript, §3 will be expanded with a new subsection providing explicit constructions of the required vectorial dual-bent functions (drawing on known scalar bent function families extended componentwise) together with a self-contained existence proof for the stated dimensions. This will directly confirm the parameters used in the codebook mapping and the resulting asymptotic optimality. revision: yes

  2. Referee: [§4] §4 (Correlation analysis): The derivation that the vectorial dual-bent property directly yields the stated maximum cross-correlation amplitude and amplitude distribution must be checked for any additional cross terms introduced by the vectorial embedding; if the reduction to the scalar bent case is not exact, the explicit formulas and asymptotic optimality do not hold.

    Authors: The original derivation relies on the vectorial dual-bent definition ensuring componentwise reduction to the scalar case. To address the concern rigorously, the revised §4 will include an additional lemma that computes the inner products explicitly and demonstrates that the vectorial embedding introduces no extra cross terms (due to the dual-bent orthogonality). This will confirm the stated maximum amplitude, full distribution, and asymptotic optimality. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions derive correlation bounds from bent-function properties without reduction to inputs or self-citations

full rationale

The paper states it constructs codebooks from vectorial dual-bent functions and explicitly determines max cross-correlation and amplitude distributions from the bent property. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation chain remains self-contained against external bent-function literature; existence assumptions are separate from circularity analysis. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the abstract or described construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the constructions rest on the existence and correlation properties of vectorial dual-bent functions drawn from prior literature; no free parameters, axioms, or invented entities are visible in the provided text.

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