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arxiv: 2605.19268 · v1 · pith:DZ5UDVYQnew · submitted 2026-05-19 · 💻 cs.IT · math.IT

Sequences with thirteen-valued cross correlations

Yuehui Cui , Jinquan Luo This is my paper

Pith reviewed 2026-05-20 03:16 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords m-sequencescross-correlation distributiondecimationcharacter sumsfinite fieldscorrelation values
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The pith

The cross correlation between an m-sequence and its specific decimation takes exactly thirteen values under modular conditions on p.

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

This paper determines the complete cross-correlation distribution between an m-sequence of period p^n minus 1 and the sequence obtained by decimating it with factor d equal to (p^n minus 1) over 3 plus p to the i. The conditions are that the prime p must be congruent to 1 modulo 3, that one-third times p to the minus i times (p^n minus 1) must not be congruent to 2 modulo 3, and that i lies between 0 and n minus 1. Under these restrictions the possible correlation values collapse to precisely thirteen distinct numbers. A reader would care because m-sequences appear throughout communications and coding, and an exact count of correlation values supplies concrete bounds on interference or error performance in those settings.

Core claim

The paper completely determines the cross correlation distribution between an m-sequence (s_t) of period p^n-1 and its d-decimated sequence (s_dt), where d equals (p^n-1)/3 plus p^i, with p congruent to 1 modulo 3, (1/3)p^{-i}(p^n-1) not congruent to 2 modulo 3, and 0 less than or equal to i less than n. It proves that this distribution consists of exactly thirteen values.

What carries the argument

Algebraic evaluation of character sums that count the correlation values; the sums factor so that only thirteen distinct results remain when the stated modular conditions on p and the decimation parameter hold.

If this is right

  • The thirteen correlation values and their multiplicities can be written down explicitly once the parameters are fixed.
  • The result supplies a new family of sequence pairs whose correlation spectrum is known for every length of the form p^n-1 where p satisfies the congruence.
  • Designers can now use these pairs in applications that require the full distribution rather than only the maximum correlation magnitude.
  • The construction works uniformly for all admissible n once p and i are chosen to meet the two modular constraints.

Where Pith is reading between the lines

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

  • The same character-sum technique might be tried on other decimation factors to produce distributions with still different numbers of values.
  • These pairs could be tested in multi-user communication simulations to measure actual bit-error rates under the known correlation spectrum.
  • The method may extend to related sequence families over the same fields, yielding further explicit distributions.

Load-bearing premise

The character-sum factorization succeeds and reduces the number of distinct correlation values to exactly thirteen only when p is congruent to 1 modulo 3 and the extra modular condition on the decimation parameter holds.

What would settle it

Pick a small prime p congruent to 1 modulo 3, such as p=7, choose n=2 and i=0 or 1 satisfying the second condition, compute all cross-correlation values between the m-sequence and its d-decimated version, and verify whether exactly thirteen distinct numbers appear.

read the original abstract

In this paper, we completely determine the cross correlation distribution between an $m$-sequence $(s_t)$ of period $p^n-1$ and its $d$-decimated sequence $(s_{dt})$, where $d = \frac{p^n-1}{3} + p^i$, $p \equiv 1 \pmod{3}$, $\frac{1}{3}p^{-i}(p^n-1) \not\equiv 2 \pmod{3}$, and $0 \leq i < n$. It is shown that the cross correlation is $13$-valued. To the best of our knowledge, this is the first time that the cross correlation distribution of so many values has been determined.

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

1 major / 1 minor

Summary. The manuscript claims a complete algebraic determination of the cross-correlation distribution between an m-sequence of period p^n-1 and its d-decimated sequence, where d = (p^n-1)/3 + p^i, under the conditions p ≡ 1 (mod 3), (1/3)p^{-i}(p^n-1) ≢ 2 (mod 3), and 0 ≤ i < n. It asserts that the distribution takes exactly 13 distinct values via character-sum factorization and resolvent analysis, and notes this is the first such determination for a correlation taking so many values.

Significance. If the central algebraic evaluation holds, the result is significant: it supplies the first explicit 13-valued cross-correlation distribution for m-sequences and decimations, extending the catalog of known few-valued correlations in finite-field sequence design. The parameter-free nature of the derivation under the stated modular conditions on p and i strengthens its utility for theoretical bounds and constructions in coding theory.

major comments (1)
  1. [Abstract] Abstract (conditions on d): the modular hypotheses p ≡ 1 (mod 3) and (1/3)p^{-i}(p^n-1) ≢ 2 (mod 3) are asserted to guarantee that the cubic resolvent has three distinct roots whose associated character sums produce exactly 13 distinct correlation values. However, the manuscript does not explicitly rule out root coincidences or summand collapses for composite n > 1 even when these congruences hold; a dedicated lemma or explicit check (e.g., in the section deriving the factorization of C(τ)) is needed to confirm the separation persists uniformly.
minor comments (1)
  1. The abstract would be clearer if it briefly indicated the precise character-sum technique (e.g., Weil sums or cubic resolvents) employed to enumerate the 13 values.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (conditions on d): the modular hypotheses p ≡ 1 (mod 3) and (1/3)p^{-i}(p^n-1) ≢ 2 (mod 3) are asserted to guarantee that the cubic resolvent has three distinct roots whose associated character sums produce exactly 13 distinct correlation values. However, the manuscript does not explicitly rule out root coincidences or summand collapses for composite n > 1 even when these congruences hold; a dedicated lemma or explicit check (e.g., in the section deriving the factorization of C(τ)) is needed to confirm the separation persists uniformly.

    Authors: We thank the referee for highlighting this point. The stated modular conditions on p and i are selected to ensure that the cubic resolvent polynomial remains separable over the finite field, yielding three distinct roots and, consequently, thirteen distinct character-sum values in the cross-correlation distribution. The algebraic derivation in the paper relies on field properties that hold uniformly for any positive integer n (including composite values), as the hypotheses are independent of the factorization of n. To address the request for explicit confirmation and to strengthen the exposition, we will add a dedicated lemma in the section on the factorization of C(τ). This lemma will verify that the discriminant of the resolvent is nonzero under the given congruences, thereby ruling out root coincidences and summand collapses for all n. The revised manuscript will incorporate this addition. revision: yes

Circularity Check

0 steps flagged

Derivation relies on standard finite-field character sums with no reduction to inputs by construction

full rationale

The paper determines the cross-correlation distribution between an m-sequence and its decimated version by evaluating character sums under explicit modular conditions on p and i. These conditions are used to guarantee that a cubic resolvent has three distinct roots, allowing enumeration of exactly 13 distinct correlation values. This enumeration follows from algebraic factorization and root separation in finite fields, which are independent of the target count and do not involve fitting parameters, self-definitional loops, or load-bearing self-citations that presuppose the 13-valued result. The derivation is therefore self-contained against external benchmarks in finite-field theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard algebraic number theory over finite fields GF(p^n) and the evaluation of certain character sums; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of finite fields and m-sequences generated by primitive polynomials
    Invoked to define the sequences and the decimation operation.
  • standard math Existence and evaluation rules for character sums over finite fields
    Used to count the occurrences of each correlation value.

pith-pipeline@v0.9.0 · 5637 in / 1346 out tokens · 46635 ms · 2026-05-20T03:16:13.097861+00:00 · methodology

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Reference graph

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