Constructions of Optimal Frequency-Hopping Sequences with Controlled Minimum Gaps

Chen Li; Chunlei Li; Dian Li; Xiangyong Zeng

REVIEW 2 major objections 3 minor 26 references

Constructions of Optimal Frequency-Hopping Sequences with Controlled Minimum Gaps

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that long optimal frequency-hopping sequences can be built by concatenating short blocks, with the long sequence's Hamming autocorrelation exactly equal to that of a short ordering sequence.

desk verdict The recursive construction in Theorem 2 is real and the central Hamming-correlation identity checks out; the paper needs only minor edge-case fixes and a cleaner proof for one inequality. read the letter →

arxiv 2506.00945 v1 pith:3AY2EXML submitted 2025-06-01 cs.IT math.IT

Chen Li , Chunlei Li , Xiangyong Zeng , Dian Li This is my paper · ORCID

classification cs.ITmath.IT MSC 94A55

keywords frequency-hoppingsequencesHammingautocorrelationwide-gapFHSminimumgapuniformoptimaldifferenceunitsetrecursiveconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's aim is to show that good frequency-hopping sequences (FHSs) with deliberately wide gaps between consecutive frequencies can be built by concatenating short blocks, and that the correlation property of the long sequence is completely controlled by a short ordering sequence. It first proves tight upper bounds on the minimum gap a uniform FHS can have. Its main result, Theorem 2, takes any permutation $\pi$ on $\{0,\dots,2m-1\}$, picks step sizes $d_1$ and $d_2$ whose common divisor with the length $\ell$ is $m$, and concatenates $2m$ arithmetic-progression blocks according to $\pi$; the resulting length-$2\ell$ sequence has Hamming autocorrelation exactly $H_{\pi_m}$, where $\pi_m=(\pi(i)\bmod m)$, and minimum gap $d_1-1$ whenever $d_1+d_2<\ell-m+2$. Corollary 1 turns this into a recursion: an optimal uniform $(2m,m,2)$-FHS used as $\pi_m$ yields optimal $(2\ell,\ell,2)$-FHSs with controlled gaps. This matters because gap control is what makes FH systems resilient to narrow-band and partial-band jamming, and the construction reduces a seemingly hard long-sequence design problem to choosing a known short optimal sequence.

What carries the argument

The load-bearing object is the concatenation-ordering sequence $\pi_m$, defined as the projection modulo $m$ of the permutation $\pi$ that orders the $2m$ row blocks. The blocks themselves are arithmetic progressions $s_j=(id_1+j)$ and $t_k=(id_2+k)$ of length $\ell/m$, where $\gcd(\ell,d_1)=\gcd(\ell,d_2)=\gcd(\ell,d_2-d_1)=m$. The identity $H_u=H_{\pi_m}$ does the work: cross-correlations between different blocks vanish, each row pair $(s_j,t_j)$ contributes exactly one Hamming agreement at every shift, so all correlation that survives in the long sequence is the correlation of the ordering sequence. The inequality $d_1+d_2<\ell-m+2$ then guarantees that the smallest jump at a concatenation boundary is still larger than the intra-block jump $d_1$, so the minimum gap is $d_1-1$.

What would settle it

A counterexample would be a valid choice of $\ell,d_1,d_2,m$ and a permutation $\pi$ with $H_{\pi_m}=2$ satisfying $d_1+d_2<\ell-m+2$ for which the sequence $u$ from Construction 1 has $H_u>2$ or minimum gap different from $d_1-1$; for instance, computing the full periodic autocorrelation of the displayed sequence in Example 3(i) or in Application I at every delay would settle the identity for those parameters.

Watch

Extended reading notes

Core claim

The central claim is the identity $H_u = H_{\pi_m}$ for sequences $u$ built in Construction 1: the periodic Hamming autocorrelation of the long sequence over $\mathbb{Z}_\ell$ is exactly the periodic Hamming autocorrelation of the short projected permutation $\pi_m$ over $\mathbb{Z}_m$. Because of this, optimizing the long sequence and optimizing the ordering sequence are the same task. The paper proves that if the short sequence $\pi_m$ is an optimal uniform FHS with parameters $(2m,m,2)$, then $u$ is an optimal uniform $(2\ell,\ell,2)$-FHS, and under the inequality $d_1+d_2<\ell-m+2$ its minimum gap is exactly $d_1-1$. The same framework also gives a direct two-block and three-block construction using difference unit sets, which reproduces and widens earlier wide-gap constructions, and it gives tight bounds showing the achieved gaps cannot generally be improved.

Load-bearing premise

The construction is only as good as its seed: Corollary 1 guarantees an optimal long sequence only when $\pi_m$ is already an optimal uniform $(2m,m,2)$-FHS, so the method inherits the existence of such short optimal sequences.

Editorial extensions

If this is right

Any optimal uniform $(2m,m,2)$-FHS can be inflated to an optimal uniform $(2\ell,\ell,2)$-FHS with minimum gap $d_1-1$ whenever $m\mid\ell$ and $d_1,d_2$ satisfy the gcd and sum conditions.
Each such short seed $\pi_m$ yields $2m$ distinct long sequences, one for each permutation $\pi$ that projects to $\pi_m$, as stated in Remark 2.
Taking $d_1=\lfloor(\ell-1)/2\rfloor$ and $d_2=d_1+1$ makes the minimum gap reach the tight upper bound from Proposition 1, so the construction attains the theoretical maximum gap for those parameters.
The earlier two-block and three-block constructions are special cases of Theorem 1: choosing $d_1,d_2,d_3$ from a difference unit set gives optimal $(2\ell,\ell,2)$- and $(3\ell,\ell,3)$-FHSs with minimum gap $\min\{d-1,\ell-d-1\}$.
Known optimal FHS families of even length can be reused as ordering sequences, turning seed sequences whose gaps were uncontrolled (sometimes 0) into wide-gap sequences, as demonstrated in Applications I through III.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A likely extension not pursued here is to use an optimal FHS set as the ordering object rather than a single sequence, producing wide-gap sequence sets; the paper only mentions this direction for no-hit-zone multi-timeslot sets.
The recursion could be iterated: since the length-$2\ell$ output has the same correlation as $\pi_m$, it could itself serve as an ordering sequence for further lengthening, though the achievable minimum gap shrinks as $m$ grows, suggesting a trade-off between length and preserved gap.
The tight gap upper bounds imply an inherent limit: for even $\ell$ with $\ell$ dividing the length, no uniform sequence can have minimum gap $\ell/2-1$, so a system that wants larger gaps must give up uniformity or use an odd-length alphabet.
A natural stress test of the identity is to use a random permutation as $\pi_m$: Theorem 2 predicts the long sequence's correlation exactly matches the random short sequence's correlation, so wide gaps do not accidentally repair poor correlation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The main thing you should know: the recursive construction in Theorem 2 is genuine and the central identity H_u = H_πm holds. I checked the part of the proof that looks shakiest—the upper bound in Case 2—and it survives. The claim that an S→T match in the first sum and a T→S match in the second cannot coexist is justified by the i0+i1 congruence argument; the same-orientation cases are actually easier because the required indices are congruent modulo ℓ1 and the index ranges are disjoint. The reverse inequality H_πm ≤ H_u is not proved cleanly—the paragraph is hand-wavy—but it follows immediately from Case 1, where τ = rℓ1 gives exact equality. So the theorem is correct, though the exposition should be tightened.

What is new: Theorem 2's lifting of an optimal uniform (2m,m,2)-FHS to length 2ℓ via a permutation of subsequences is a genuinely different construction device from the earlier (d,d+1,d+2) work in [21]. Theorem 1 generalizes that work to arbitrary difference-unit-set multipliers, which is a moderate but useful extension. The applications to sequences from [8,9,12] and the worked examples give the paper practical value.

Soft spots, in proportion: - Proposition 1 is false for ℓ=2, n even: the uniform sequence (0,1,0,1) has minimum gap 0 while the stated 'otherwise' bound is −1. This is an edge-case oversight; it should be flagged with an exception, and it doesn't touch the constructions. - Remark 2 says each π_m lifts to 2m permutations; it should be 2^m. Cosmetic. - The proof of H_πm ≤ H_u should be rewritten as a one-liner using Case 1, and Case 2's upper bound needs a bit more room to explain the orientation handling.

Citation pattern looks fair; the differences with [21] and [26] are clearly credited.

Who it's for: people working on FHS design and Hamming-correlation-optimal sequences. They'll get a new recursive tool and a clean way to add minimum-gap control. The paper deserves a serious referee. I'd send it out, asking for the small fixes and a clearer proof of the reverse inequality.

Referee Report

2 major / 3 minor

Summary. The manuscript studies uniform frequency-hopping sequences over Z_l with optimal Hamming correlation and controlled minimum gaps. Section 3.1 states upper bounds on the minimum gap of a uniform FHS and claims these bounds are tight. Section 3.2 gives a first construction, generalizing results of Li et al. [21] by replacing consecutive differences with arbitrary elements of a difference unit set, and proves optimality of the resulting (2l,l,2)- and (3l,l,3)-FHSs. Section 3.3 presents the main recursive construction (Construction 1): under gcd(l,d1)=gcd(l,d2)=gcd(l,d2-d1)=m, the rows of two m by l/m matrices are concatenated in an order specified by a permutation pi, and Theorem 2 asserts that the Hamming correlation of the resulting sequence u equals that of pi_m=(pi(i) mod m), with minimum gap d1-1 when d1+d2<l-m+2. Corollary 1 then converts an optimal uniform (2m,m,2)-FHS pi_m into an optimal (2l,l,2)-WGFHS. Section 3.4 applies the construction to optimal FHSs from [8], [9], and [12].

Significance. If the main construction is sound, the paper gives a useful reduction: optimal long wide-gap FHSs are obtained from optimal short uniform FHSs by an explicit interleaving scheme with controlled minimum gaps. The construction is parameter-free in the sense that no fitted or data-dependent quantities are used, and the Hamming-correlation identity is supported by lemmas on pairwise subsequence correlations rather than by numerical fitting. The generalization of [21] to difference-unit-set elements is of interest to researchers in frequency-hopping sequence design. However, the bounds section contains false edge cases, and the proof of the central identity in Theorem 2 is too compressed in places; these issues need to be fixed before the claims can be accepted.

major comments (2)

[Section 3.1, Proposition 1] Proposition 1 is false as stated for very short sequences. For n=2 over Z_3, the uniform sequence (0,2) has minimum gap min{|2-0|,|0-2|}-1=1, while the 'otherwise' bound is floor((3-1)/2)-1=0. For n=2 over Z_4, the uniform sequence (0,3) has minimum gap 2, while the first-case bound is l/2-1=1. For l=2 and n=4, the uniform sequence (0,1,0,1) has minimum gap 0, while the 'otherwise' bound is -1. The proof treats only n=ql±r with a positive integer q and therefore does not cover n=2. Please restate the proposition with an explicit exception, or give a separate bound for n<=2, and re-verify the reachability claim.
[Section 3.3, proof of Theorem 2, Case 2] The upper-bound half of the key identity Hu=Hpi_m is not proved by a complete case analysis as written. After the reduction to at most 2h matches for the shifts r-1 and r, the text derives an inequality for one mixed orientation (h[s,t] in the first summation range versus h[t,s] in the second) and says the other orientation is similar; it does not explicitly dispose of the same-orientation cases or justify the arbitrary pairing between the kappa_v and kappa'_v lists. A complete proof should show, for fixed tau_1, that a matching index in the first range and a matching index in the second range cannot both exist regardless of orientation and superscript. The subsequent paragraph proving Hpi_m <= Hu is also heuristic; a rigorous argument follows from equation (10) at delays tau = r l_1, which gives Hu(r l_1) = Hpi_m(r), combined with the symmetry Hpi_m(r) = Hpi_m(2m-r). Because Corollary 1 converts this identity into optimality of the constructed sequences, this part needs to be expanded and made fully formal.

minor comments (3)

[Section 3.3, Remark 2 and Section 3.4, Application I] Remark 2 states that a uniform length-2m sequence pi_m has 2m lifts to a permutation pi, but the correct number is 2^m: for each of the m residues, the two positions carrying that residue can be assigned the two preimages r and r+m in either order. Example 3 itself lists 8=2^3 permutations for m=3, which contradicts the stated 2m=6. The same error appears in Application I, where '2N different permutations' should read '2^N different permutations.'
[Section 3.3, proof of Theorem 2, final paragraph] The paragraph proving Hpi_m <= Hu is confusing because it appears to discuss pointwise delays tau in the long sequence u while Hpi_m is only defined for delays in {1,...,2m-1}. Replacing this paragraph with the direct argument using equation (10) at delays r l_1 would make the proof shorter and clearer.
[Section 3.1, proof of Proposition 1] The proof of the even-l, n even, l not dividing n case writes n=ql+r with q a positive integer and 2<=r<=l/2; this excludes n=2, which is the source of the counterexamples noted above. Please add an explicit treatment of small n or state the intended domain of the proposition.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Hamming-correlation identity and minimum-gap results are proven from explicit constructions and external, non-overlapping prior results.

full rationale

The paper's central claim is Theorem 2, asserting that the sequence u from Construction 1 has Hamming correlation equal to that of the derived ordering sequence πm, and minimum gap d1−1 under the stated inequality. This is not circular: u is explicitly constructed from length-l1 rows sj and tk, Lemma 3 proves the needed subsequence correlation facts from the gcd conditions, and the proof of Hu = Hπm is a case analysis on the delays rather than an assumption or a fit. Corollary 1 transfers optimality from an external optimal uniform (2m,m,2)-FHS πm; this is a genuine modular reduction, with the seed supplied by independent constructions in [8], [9], and [12]. No parameter is fitted to any subset of the output data, and no prediction is equal to an input by construction. The minimum-gap analysis is a direct computation of adjacent differences in the concatenation. The cited prior work [21] is cited as background and as a special case, not as the justification of the new theorem, and none of the authors overlap with the sources [8], [9], [12], [18], or [26] in a way that makes the load-bearing evidence self-referential. The skeptic's concern about the completeness of the Case 2 argument in Theorem 2 and the edge case in Proposition 1 are proof-correctness issues, not circularity: even if those arguments needed repair, the reduction would still not be circular. The derivation is self-contained modulo standard external facts and established optimal-sequence families. Score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical constants are fitted to data. The parameters l, d1, d2, d3, m and the permutation pi are free design choices, so the free-parameter count is zero. The construction relies on standard bounds and on cited external results about difference unit sets and known optimal FHS families; these are entered as axioms rather than hidden assumptions.

assumptions (5)

standard math Lempel-Greenberger lower bound on Hamming autocorrelation (Lemma 1)
Used to define optimality of FHSs throughout; stated and cited to [2].
domain assumption Wide-gap bound of Li et al. (Lemma 2)
Used for comparison of bounds in Section 3.1; cited to [18].
domain assumption Difference unit set cross-correlation property (Proposition 4)
Load-bearing for Theorem 1: any two distinct decimated sequences from DU(Z_l) have Hamming cross-correlation exactly 1; cited to [26].
domain assumption Existence of optimal uniform (2m,m,2)-FHS seeds
Corollary 1 requires an optimal uniform short sequence pi_m; the paper supplies examples from [8,9,12].
standard math Uniform FHSs are the only candidates for optimal sequences under the Lempel-Greenberger bound
Used in Corollary 1 when passing from Hu=Hpi_m to optimality; follows from the bound derivation.

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Cite this review

Pith. "Pith review of Constructions of Optimal Frequency-Hopping Sequences with Controlled Minimum Gaps." pith.science (2026). https://pith.science/paper/3AY2EXML

@misc{pith2026250600945,
  author       = {Pith},
  title        = {Pith review of: Constructions of Optimal Frequency-Hopping Sequences with Controlled Minimum Gaps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AY2EXML}},
  note         = {Machine review of arXiv:2506.00945}
}

read the original abstract

Frequency-hopping sequences (FHSs) with low Hamming correlation and wide gaps significantly contribute to the anti-interference performance in FH communication systems. This paper investigates FHSs with optimal Hamming correlation and controlled minimum gaps. We start with the discussion of the upper bounds on the minimum gaps of uniform FHSs and then propose a general construction of optimal uniform wide-gap FHSs with length 2l and 3l, which includes the work by Li et al. in IEEE Trans. Inf. Theory, vol. 68, no. 1, 2022 as a special case. Furthermore, we present a recursive construction of FHSs with length 2l, which concatenate shorter sequences of known minimum gaps. It is shown that the resulting FHSs have the same Hamming correlation as the concatenation-ordering sequences. As applications, several known optimal FHSs are used to produce optimal FHSs with controlled minimum gaps.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [21]

Constructions of optimal uniform wide-gap frequency-hopping sequences,

P. Li, C. Fan, S. Mesnager, Y. Yang, and Z. Zhou, “Constructions of optimal uniform wide-gap frequency-hopping sequences,” IEEE Trans. Inf. Theory , vol. 68, no. 1, pp. 692–700, Jan. 2022

work page 2022

[2] [26]

Combinatorial characterizations of one-coincidence frequency-hopping sequences,

Z. Cao, G. Ge, and Y. Miao, “Combinatorial characterizations of one-coincidence frequency-hopping sequences,” Des., Codes Cryptogr. , vol. 41, no. 2, pp. 177–184, Nov. 2006. 26

work page 2006

[3] [8]

New classes of optimal frequency-hopping sequences by interleaving techniques,

J.-H. Chung, Y. K. Han, and K. Yang, “New classes of optimal frequency-hopping sequences by interleaving techniques,” IEEE Trans. Inf. Theory , vol. 55, no. 12, pp. 5783–5791, Dec. 2009

work page 2009

[4] [9]

Optimal frequency-hopping sequences with new parame- ters,

J.-H. Chung and K. Yang, “Optimal frequency-hopping sequences with new parame- ters,” IEEE Trans. Inf. Theory , vol. 56, no. 4, pp. 1685–1693, Apr. 2010

work page 2010

[5] [12]

Sets of optimal frequency-hopping sequences,

C. Ding and J. Yin, “Sets of optimal frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3741–3745, Aug. 2008

work page 2008

[6] [1]

M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread spectrum commu- nications handbook. New York: McGraw-Hill, 1994

work page 1994

[7] [2]

Families of sequences with optimal Hamming- correlation properties,

A. Lempel and H. Greenberger, “Families of sequences with optimal Hamming- correlation properties,” IEEE Trans. Inf. Theory, vol. 20, no. 1, pp. 90–94, Jan. 1974

work page 1974

[8] [3]

Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences,

D. Peng and P. Fan, “Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences,” IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 2149–2154, Sep. 2004

work page 2004