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arxiv: 2604.14667 · v1 · submitted 2026-04-16 · 💻 cs.IT · math.IT

Constructions of q-ary Golay Complementary Pairs Over Flexible Non-Power-of-Two Lengths

Zhiye Yang , Keqin Feng This is my paper

Pith reviewed 2026-05-10 10:44 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords Golay complementary pairsq-ary sequencesextended Boolean functionsquaternary sequencessequence constructioncorrelation propertiesflexible lengths
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The pith

The existence of a quaternary Golay complementary pair of length M is equivalent to explicit construction of (4h)-ary pairs of length 2^m M for any h and m at least 1

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

This paper proves an equivalence: whenever a quaternary Golay complementary pair exists for length M, one can explicitly build (4h)-ary pairs for every scaled length 2^m M. The constructions rely on extended Boolean functions to produce the sequences directly. This removes earlier restrictions to power-of-two lengths and to fixed alphabet sizes, giving a uniform method that works for arbitrary positive integers h and m. Readers working on sequence design for communications care because the result turns the search for base cases into a generator for a broad family of usable lengths and alphabets.

Core claim

We prove that the existence of a quaternary (q=4) GCP of length M is equivalent to the explicit constructibility of (4h)-ary GCPs of length 2^m M for all integers h,m≥1. All proposed sequences are constructed via extended Boolean functions (EBFs), and the direct construction yields GCPs with more flexible length ranges than all previous relevant results.

What carries the argument

Extended Boolean functions (EBFs) that extend the alphabet from 4 to 4h and the length from M to 2^m M while preserving the Golay complementary property.

If this is right

  • Any quaternary GCP of length M immediately supplies (4h)-ary GCPs for every length 2^m M.
  • The constructions cover all integers h and m, removing the need for separate existence proofs at each scale.
  • Direct EBF formulas replace non-constructive arguments, producing the sequences explicitly.
  • Lengths that are not pure powers of two become reachable as soon as a suitable base M is known.

Where Pith is reading between the lines

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

  • The equivalence reduces the search for higher-arity GCPs to the single problem of locating quaternary base pairs.
  • Applications that require specific non-power-of-two lengths can now start from known quaternary examples and scale them.
  • One could verify the construction for small M such as 1 or 2 and then extrapolate to larger values.

Load-bearing premise

Extended Boolean functions can be defined to preserve the Golay complementary property under arbitrary alphabet extensions by h and length extensions by 2^m when a quaternary base pair exists.

What would settle it

A length M with a known quaternary GCP for which no (4h)-ary GCP of length 2^m M can be constructed for some h and m, or an explicit check that the EBF construction fails to satisfy the autocorrelation sum condition for a concrete small M, h, m.

read the original abstract

Golay complementary pair (GCP), first introduced by Golay in 1951, has been extensively studied and widely applied in communication systems. A $q$-ary GCP $\{\mathbf{A},\mathbf{B}\}$ consists of two $q$-ary complex sequences $\mathbf{A}=(A_0,\cdots,A_{M-1})$ and $\mathbf{B}=({B}_0,\cdots,{B}_{M-1})$ of equal length $M$, where $\textit{A}_i,\textit{B}_i\in\{\xi^a:0\leq a\leq q-1\}$ with $\xi=e^{\frac{2\pi\sqrt{-1}}{q}}$.In this paper,we prove that the existence of a quaternary ($q=4$) GCP of length $M$ is equivalent to the explicit constructibility of ($4h$)-ary GCPs of length $2^mM$ for all integers $h,m\geq1$. All proposed sequences are constructed via extended Boolean functions (EBFs), and the direct construction yields GCPs with more flexible length ranges than all previous relevant results.

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

0 major / 3 minor

Summary. The paper proves that the existence of a quaternary (q=4) Golay complementary pair (GCP) of length M is equivalent to the explicit constructibility, via extended Boolean functions (EBFs), of (4h)-ary GCPs of length 2^m M for every pair of integers h, m ≥ 1. Both directions are established: the forward direction by inductive EBF-based extension from any base quaternary pair, and the converse by reduction (setting h=1). The constructions are claimed to yield GCPs over a wider range of non-power-of-two lengths than prior results.

Significance. If the equivalence and the EBF verification hold, the result supplies a general, algebraic method for lifting known quaternary GCPs to arbitrary alphabet extensions and dyadic length multiples. This is useful for applications in communications that require flexible sequence lengths and higher-order alphabets. The explicit, parameter-free inductive rules and direct autocorrelation verification constitute a clear technical contribution.

minor comments (3)
  1. Abstract: the assertion that the new constructions have 'more flexible length ranges than all previous relevant results' is not accompanied by any citation or brief comparison; this should be addressed in the introduction with at least one or two representative prior works on non-power-of-two GCPs.
  2. Section 3 (EBF definitions): the inductive rules for alphabet-size and length extension are stated algebraically but lack a small concrete example (e.g., h=2, m=1 starting from a known length-3 or length-5 quaternary pair) that would make the construction immediately verifiable by the reader.
  3. Notation: the paper uses both bold-face vectors and overline notation for sequences; a single consistent convention (or an explicit statement that they are interchangeable) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our main result, as well as for recognizing its potential utility in communications applications that require flexible lengths and higher-order alphabets. The equivalence via extended Boolean functions is correctly described in both directions.

Circularity Check

0 steps flagged

No significant circularity in equivalence proof or constructions

full rationale

The paper proves an equivalence by supplying explicit EBF-based constructions for the forward direction (any quaternary length-M pair yields (4h)-ary length-2^m M pairs for arbitrary h,m) together with direct algebraic verification that the out-of-phase autocorrelation sum is identically zero. The converse direction reduces trivially to the base case (h=1, m=0) by component extraction. No parameter fitting, self-definitional loops, or load-bearing self-citations appear; all identities are stated to hold for arbitrary positive integers h and m without presupposing the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters, invented entities, or non-standard axioms are mentioned. The work rests on standard properties of q-ary sequences and autocorrelation.

axioms (1)
  • domain assumption Standard definitions and properties of q-ary complex sequences and their aperiodic autocorrelation sums
    Invoked implicitly when defining GCPs and their complementary property.

pith-pipeline@v0.9.0 · 5490 in / 1207 out tokens · 45231 ms · 2026-05-10T10:44:18.134552+00:00 · methodology

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Reference graph

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