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Correcting Bursty/Localized Deletions: A New Error-Position-Estimation Code

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Pith's one-line read For any fixed alphabet size q and any burst length t < 2q (with q even in the upper range), there are q-ary codes correcting a burst of at most t deletions—and codes correcting a single t-localized deletion—with redundancy log n + (t-1)…

desk verdict A genuinely new position-estimation scheme that improves redundancy for burst and localized deletions, but the main theorems currently rest on an unstated parameter regime of a cited lemma; fixable, but unsupported as written. read the letter →

arxiv 2507.04797 v1 pith:W3I2WGU2 submitted 2025-07-07 cs.IT math.IT

Zuo Ye , Yubo Sun , Gennian Ge This is my paper · ORCID
classification cs.ITmath.IT MSC 94B6094B35
keywords burst-deletioncorrectingcodeslocalized-deletionposition-estimationcodelocally-balancedsequencesdifferentialVarshamov-Tenengoltsconstraintq-aryredundancybounds
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

Deletions that hit a sequence in one burst, or inside one short window, are hard because the receiver does not know where symbols vanished. This paper constructs codes that fix at most t such deletions while adding only about log n + (t-1) log log n + O(1) redundant symbols over an alphabet of size q, for any fixed q and any 2 ≤ t < 2q (with q even when t ≥ q). That improves the redundancy of previous constructions for both the burst-deletion and localized-deletion models, and it comes from a new position-estimation code: codewords are chosen so that a short checksum-driven invariant pinpoints the damaged window before correction begins. The paper also provides an efficient encoder that maps an arbitrary input into a codeword whose differential sequence is strong-(ℓ,ϵ)-locally-balanced, the property the whole construction relies on.

What carries the argument

The machinery is the differential sequence ψ(x), defined by ψ(x)_i = (x_{i−1} − x_i) mod q with x_0 = x_{n+1} = 0; deleting one symbol simply merges two adjacent entries of ψ with a mod-q sum. The codes select codewords whose ψ is strong-(ℓ,ϵ)-locally-balanced—every substring of length at least ℓ has L1-weight near (q−1)/2 per symbol—and then impose the two checksum constraints on ψ. The locally-balanced condition makes the key gap equation (j − i)Δsum + σ(j) − σ(i) impossible when j − i ≥ ℓ, unless a 'good triple' (q,t,ϵ) with a certain integer s_{t′} exists; Lemma III.4 shows such triples exist exactly for t < q or even q with t < 2q. The final correction uses a P-bounded (t1,t2)-burst-error correcting code from Lemma III.3 to repair whatever remains in a window of length P = ℓ + t − 1.

What would settle it

Run a finite computer search for q=3, t=2, and small n, enumerating all codewords satisfying the constraints in Theorem IV.1 and testing whether any two have overlapping two-deletion-burst balls; a single intersection would refute the construction in that parameter range. A narrower check is to verify whether Lemma III.3 actually supplies the t2=0 case, since the final correction step in Theorem IV.1 (and Theorem V.1 at t′=t) depends on that unstated case.

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Extended reading notes

Core claim

The central claim is Theorem IV.1 and Theorem V.1: for fixed q and t with 2 ≤ t < 2q (even q required in the upper range), there is a q-ary code correcting a burst of at most t deletions, and a q-ary code correcting a single t-localized deletion, both with redundancy log n + (t-1) log log n + O(1). The codes are intersections of four constraints: the differential sequence ψ(x) is strong-(ℓ,ϵ)-locally-balanced; for each possible burst length t′ a P-bounded code checksum is fixed; VT(ψ(x)) ≡ b (mod N); and Sum(ψ(x)) ≡ cq (mod (t+1)q). The VT and L1 constraints carry the position information that survives deletion, and the locally-balanced condition bounds how far wrong the position estimate can be—Claim 1 shows the estimated start satisfies j − i < ℓ, a window of length O(log n). Inside that window a P-bounded burst-error code finishes the correction. Because the same position-estimation code handles every burst length t′ and also corrects a single deletion on its own, the redundancy saves one log log n factor compared with approaches that split the alphabet or handle each burst length separately.

Load-bearing premise

The decoding proof relies on a component that fixes purely deleted symbols inside a known short window, but the lemma it cites is stated only for mixed deletion-and-substitution errors, and the purely deletion case is not proved separately.

Editorial extensions

If this is right

  • For q-ary alphabets with t < q, the (≤t)-burst code achieves redundancy log n + (t−1) log log n + O(1), improving on the previous log n + 8 log log n + o(log log n) for general q.
  • For even q in the range q ≤ t < 2q, both the burst and localized codes achieve the same redundancy, and the localized code improves on the previous log n + 2t log log n + O(1).
  • The position-estimation code corrects a single deletion on its own and serves all burst lengths up to t with one set of constraints, so the redundancy does not accumulate an extra log log n per possible burst length.
  • The new encoder turns any length-(n−2) sequence into a length-n sequence whose differential sequence is strong-(ℓ,ϵ)-locally-balanced with two redundant symbols, running in O(n^C) time for a constant C depending on the chosen parameters.
  • The decoding algorithms run in O(n log n) time and locate the damaged window before applying the local correction step.
  • The redundancy gap to the log n + Ω(1) lower bound is narrowed but not closed; the paper leaves open whether the lower bound is asymptotically tight for general t and q.

Reading between the lines

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

  • The t < 2q restriction is an artifact of the good-triple condition; extending past t ≥ 2q would need a different local-balance ratio or a different invariant, and nothing here suggests the log-log factor is removable in that range.
  • Because ψ is a bijection onto sequences with Sum ≡ 0 (mod q), the same position-estimation scheme may transfer to insertion or edit-burst models where differential sequences also merge or split locally.
  • The encoder's two-symbol overhead raises the natural question of whether one redundant symbol suffices; the counting in Lemma III.2 gives at least q^n/2 valid codewords, so the existential bound does not rule out a one-symbol encoder.
  • The good-triple characterization in Lemma III.4 is a self-contained design tool that could be reused for other constrained codes that require local balance on differential sequences.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper proposes new q-ary codes correcting bursts of at most t deletions and t-localized deletions, achieving redundancy log n + (t−1) log log n + O(1) for q ≥ 2, t < q (or even q, t < 2q). The construction selects codewords whose differential sequences are strong-(ℓ,ε)-locally-balanced and satisfies a VT-type constraint and an L1-weight constraint. The error-correction algorithm first estimates error positions within a short window, then invokes a P-bounded (t1,t2)-burst-error correcting code on the located window. The paper also provides an efficient encoder mapping arbitrary length-(n−2) sequences to length-n sequences with strong locally balanced differential sequences. The main technical novelty is a simpler position-estimation code that also corrects a single deletion.

Significance. If the technical gaps identified below are resolved, the redundancy improvement is real and meaningful: it saves one log log n factor for all q with t<q and extends to even q up to t<2q, and the position-estimation method is conceptually simpler than prior array-based or dense-sequence methods. The efficient encoder for strong locally balanced differential sequences appears to be new. The paper is well-structured and contains detailed proof sketches for the main lemmas, with external results from [19] and [31] used as black boxes.

major comments (2)
  1. [Lemma III.3; Theorems IV.1 and V.1] Theorems IV.1 and V.1 invoke the function f_{P,t1,t2} of Lemma III.3 with t2=0 (and in Theorem V.1 also with t2=1). Specifically, Theorem IV.1 uses f_{P,t',0} for 2≤t'≤t, and Theorem V.1 uses f_{P,t,t−t'} with t−t'=0 for t'=t and t−t'=1 for t'=t−1. However, Lemma III.3 is stated only for integers t1≥t2≥2, and Definition III.2 also requires 1≤t2≤t1. No derivation is given for the cases t2=0 or t2=1. Consequently, the assertion 'By Lemma III.3, the code is a P-bounded ... code' in the final correction step of both theorems is not supported by the quoted lemma. The authors need to either extend Lemma III.3 to cover t2=0 and t2=1, or provide an alternative P-bounded code for deletion-only and single-symbol-replacement errors inside the located window, together with a proof.
  2. [Theorem V.1, Step 3] After Claim 2, the proof states that the substring y'_{[j−ℓ+1,j]} contains y'_{i_s−∑_{r=1}^{s-1} t_r} for all 1≤s≤k. This does not follow from Claim 2. Claim 2 only gives j−i1 < ℓ−t'+t1, which yields an upper bound on j and implies j−ℓ+1 < i1, but it gives no lower bound on j. To cover the starts of all deleted blocks, one needs j ≥ i_k−∑_{r=1}^{k−1} t_r, which can be as large as i1+t−t'. Since j is chosen as the first (largest) index from the right satisfying (28), the proof must show that some such index exists in that range. As written, the located window could end before the later deleted blocks, and the subsequent correction step would fail. Please provide an argument that the scanning procedure yields a window covering all block starts.
minor comments (4)
  1. [Definition III.1] The upper endpoint of the interval in Definition III.1 is written as (q+1)/2 + ε, but Lemma III.2, Claim 1, and Section VI all use (q−1)/2 + ε. Please correct this typo for consistency.
  2. [Section VI] Propositions VI.1–VI.3 are stated without proofs, with the remark that they are direct q-ary generalizations of binary results. Since these propositions are central to the claimed encoder, please provide proofs or precise references to the q-ary versions.
  3. [Theorem IV.1] The existence of parameters a_{t'}, b, c for which the redundancy bound holds is asserted without the standard counting argument. A short pigeonhole argument would make the redundancy claim complete.
  4. [Section VI-B] There is a typo in 'the anaysis'; it should be 'the analysis'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the position-estimation and redundancy derivation are self-contained; the t2=0/1 application of Lemma III.3 is a correctness gap, not a circular step.

full rationale

The main construction selects codewords whose differential sequence is strong-(l,e)-locally-balanced and then imposes VT and L1-sum constraints; the decoding algorithm recovers the syndrome values from these constraints and locates a short window via the locally-balanced property (Claims 1 and 2). The redundancy count is direct: one bit for the balance property, about (t-1) log log n for the P-bounded constraints, log n for the VT constraint, and O(1) for the sum constraint. No fitted parameter is renamed as a prediction, and no defining equation of the code is equivalent to the error-correcting property being proved. The cited Lemma III.3, from [19, Corollary 3], is a published parameter-free result by partially overlapping authors, but it does not assume the target redundancy bound and its proof is independent; therefore it does not create circularity. The serious defect is that Theorems IV.1 and V.1 invoke f_{P,t',0} and f_{P,t,t-t'} with second parameter 0 or 1, while Lemma III.3 is stated only for t1 >= t2 >= 2 and Definition III.2 requires 1 <= t2 <= t1. In the differential sequence a t'-burst deletion is a (t'+1,1)-burst-error, so the t2=0 and t2=1 cases require a separate derivation. This is a load-bearing gap in the proof, but it is a correctness gap rather than a circularity: the claimed code existence is not obtained by assuming the theorem, and the derivation does not reduce to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The construction rests on a hand-chosen balance parameter ϵ from the good-triple condition, on the external P-bounded burst-error code of [19], on a claimed direct generalization of [31], and on standard probabilistic inequalities. No new entities are introduced and no constants are fitted to data.

free parameters (2)
  • epsilon (ϵ) = Any value satisfying Lemma III.4, e.g. 0 < ϵ < min{q/(2t), q/t2 - 1/2, 1/(2(q+1))} depending on the case
    The good-triple inequalities (7) are used in Claim 1 and Claim 2 to force j-i < ℓ. The redundancy bound is independent of the exact value, so ϵ is an existence parameter rather than a fitted constant.
  • eta1, eta2, s = Chosen so that 0 < η2 < ϵ < (q-1)/2 and the inequality in Lemma VI.1 holds; s ≥ 1
    These parameters control the encoder. They must satisfy a parameterized inequality so that local balance at window ℓ1 implies strong local balance at window ℓ; no data are involved.
assumptions (4)
  • ad hoc to paper Good triple condition (Definition III.3, Lemma III.4)
    The entire construction is conditional on the existence of an integer st' in the interval (7) for every 2 ≤ t' ≤ t. This condition is introduced specifically to make the contradiction in Claim 1 go through, and it restricts q and t, e.g., t < 2q and even q when t ≥ q.
  • domain assumption P-bounded (t1,t2)-burst-error correcting codes exist with redundancy log P + O(1) (Lemma III.3 from [19, Corollary 3])
    The decoding step corrects errors inside a located window of length P using these codes. The lemma is quoted from prior work and is not proved in this paper; moreover it is stated for t2 ≥ 2 but applied with t2 = 0.
  • domain assumption Generalization of the sliding-window encoder of [31] to q-ary alphabets (Propositions VI.1-VI.3)
    The encoder's Stage 1 relies on this generalization; proofs are omitted and the paper says 'it is easy to see'. If the generalization fails, the encoder claim is unsupported.
  • standard math Hoeffding's inequality and inclusion-exclusion (Lemma III.2)
    Used to show that a positive fraction of sequences have strong-locally-balanced differential sequences.

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Pith. "Pith review of Correcting Bursty/Localized Deletions: A New Error-Position-Estimation Code." pith.science (2026). https://pith.science/paper/W3I2WGU2

@misc{pith2026250704797,
  author       = {Pith},
  title        = {Pith review of: Correcting Bursty/Localized Deletions: A New Error-Position-Estimation Code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W3I2WGU2}},
  note         = {Machine review of arXiv:2507.04797}
}
abstract

Codes correcting bursts of deletions and localized deletions have garnered significant research interest in recent years. One of the primary objectives is to construct codes with minimal redundancy. Currently, the best known constructions of $q$-ary codes correcting a burst of at most $t$ deletions ($(\le t)$-burst-deletion correcting codes) achieve redundancy $\log n+8\log\log n+o(\log\log n)$ (for any $q$ and $t$) or $\log n+t\log\log n+O(1)$ (for even $q$). For codes correcting single $t$-localized-deletion ($t$-localized-deletion correcting codes), state-of-the-art constructions attain redundancy $\log n+O\parenv{t(\log\log n)^2}$ (for any $q$ and $t$) or $\log n+2t\log\log n+O(1)$ (for even $q$). Here, $n$ denotes the code-length, and $q$ and $t$ are fixed. These codes employ a position-estimation component to approximate error positions, augmented by additional constraints that enable error-correction given the information about error positions. In this work, we select codewords from the set of sequences whose differential sequences are strong-$(\ell,\epsilon)$-locally-balanced. By imposing a VT-type constraint and an $L_1$-weight constraint on the differential sequences of codewords, we construct novel position-estimation codes. When $q\ge 2$ and $t<q$, or $q$ is even and $t<2q$, this approach gives a $q$-ary $(\le t)$-burst-deletion correcting code and a $t$-localized-deletion correcting code with redundancy $\log n+(t-1)\log\log n+O(1)$. In addition to improving previous redundancy, the method is new and our position-estimation codes are simpler than those in previous works. Finally, we give an efficient encoder to encode an arbitrary input sequence into a sequence whose differential sequence is strong-$(\ell,\epsilon)$-locally-balanced. To our knowledge, no prior algorithm for this specific task has been reported.

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