Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions

Chang-An Zhao; Huachao Zhang

REVIEW 4 major objections 5 minor 24 references

Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that all pure gaps at many totally ramified places of a Kummer extension reduce to a finite bottom set, and turns this into a recursive floor-sum algorithm that yields a length-record [74,60,≥10] code over…

desk verdict Core pure-gap characterization is real and checks out; the headline record code and Algorithm 3 need fixing before the paper is trusted. read the letter →

arxiv 2505.23274 v1 pith:BWKFHY4W submitted 2025-05-29 cs.IT math.IT

Huachao Zhang , Chang-An Zhao This is my paper

classification cs.ITmath.IT MSC 94B2711G2014H55

keywords KummerextensionspuregapstotallyramifiedplacesalgebraicgeometrycodesWeierstrasssemigroupmulti-pointbottomsetminimumdistance

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

This paper studies pure gaps—tuples of integers that are gaps at every one of several distinguished places—for Kummer extensions $y^m=\prod_{i=1}^r(x-\alpha_i)^{\lambda_i}$ over finite fields with $\gcd(m,q)=1$. It proves that at any collection of totally ramified places, the full set of pure gaps is determined by a finite bottom set inside $[1,m-1]^{s+1}$: every pure gap is a bottom tuple plus multiples of $m$, subject to one budget inequality on the sum of the multipliers (Theorem 3.1). Based on this, Algorithms 1–3 compute the bottom set recursively from one place to many, and Theorem 3.6 shows that a new pure gap can be certified by checking a single floor-sum inequality. For the special case where all exponents $\lambda_i$ are equal, the paper gives closed-form descriptions of the pure-gap sets at several places, including the infinite place, which had been open. These gaps are used to construct multi-point algebraic geometry codes with improved minimum distance; in particular a $[74,60,\geq 10]$ code over $\mathbb{F}_{25}$ has longer length than the best comparable code previously known.

What carries the argument

The central object is the bottom set $\underline{G}^0(Q_{j_0},\ldots,Q_{j_s})=G^0\cap[1,m-1]^{s+1}$: the finitely many pure gaps with all coordinates between 1 and $m-1$. The machinery is the floor-sum criterion of Lemma 3.2: a tuple $(a_0,\ldots,a_s)$ is a pure gap iff for every $0\leq v\leq s$, $$\sum_{i=0}^s\left\lfloor\frac{a_i-a_v\sigma_{j_v}\lambda_{j_i}}{m}\right\rfloor+\sum_{i=s+1}^r\left\lfloor\frac{-a_v\sigma_{j_v}\lambda_{j_i}}{m}\right\rfloor\leq -1,$$ where $\sigma_{j_v}$ is the inverse of $\lambda_{j_v}$ modulo $m$. Theorem 3.1 turns those inequalities into a bound $k_a$ on how many multiples of $m$ each coordinate may carry, so the infinite pure-gap set is compressed into the finite bottom set plus a bounded box of $m$-translates. Algorithms 1–3 implement the recursion from one place to many, and Theorem 3.6 adds a one-inequality certificate for moving from a known pure gap to a consecutive one.

What would settle it

For the record claim, recompute the displayed 18-tuple pure-gap set in Example 5.1 by applying the paper's own Lemma 3.2 to the function field $y^8=(x+1)^3(x^2+x+2)^7$; if any tuple fails the floor-sum inequalities, or if the resulting $[74,60,\geq10]$ code has true minimum distance at most 9, the claimed record collapses. For Theorem 3.1 itself, exhaustively compute Riemann–Roch dimensions for a small Kummer extension and compare the true pure-gap set with the output of Algorithm 3.

Watch

Extended reading notes

Core claim

The central claim is that, for $s+1$ totally ramified places and any Kummer extension with $\gcd(m,q)=1$, $$$G^{0}$(Q_{j_0},\ldots,Q_{j_s})=\{(mk_0+a_0,\ldots,mk_s+a_s): a\in $G^{0}$\cap[1,m-1]^{s+1},\ k_i\geq0,\ \textstyle\sum_{i=0}^s k_i\leq k_a\},$$ where $k_a$ is one integer defined by a maximum of floor sums involving the inverses $\sigma_{j_v}$ of the $\lambda_{j_v}$ modulo $m$. The proof runs through Lemma 3.2, an arithmetic criterion expressing pure-gap status as the conjunction of $s+1$ floor-sum inequalities; Theorem 3.1 then shows these inequalities are equivalent to the single budget constraint $\sum k_i\leq k_a$. In the equal-exponent case it also describes $G^0$ explicitly, for places $Q_1,\ldots,Q_s$ and for $Q_\infty,Q_1,\ldots,Q_s$, and reads off consecutive pure gaps for code construction.

Load-bearing premise

The load-bearing premise is the external floor-sum characterization (Theorem 2.7 in [6], restated as Lemma 3.2): a tuple is a pure gap exactly when certain floor sums are at most −1, which presumes every chosen place is totally ramified and rational, $\gcd(m,q)=1$, and pole divisors of $x-\alpha_i$ have the stated form; because the paper does not re-derive this criterion, all later theorems and the record code inherit it.

Editorial extensions

If this is right

For any Kummer extension with $\gcd(m,q)=1$, all pure gaps at any chosen totally ramified places can be listed by an explicit finite algorithm, removing the need to handle each function field separately.
The closed-form equal-exponent descriptions cover $Q_\infty$ as well as finite places, so multi-point code constructions can use the infinite place in the same framework.
Consecutive pure gaps obtained from Propositions 4.6, 4.8, and 4.12 feed Theorem 2.2 to improve the minimum-distance lower bound; tables in Examples 5.5–5.7 show gains over the standard comparison tables.
The paper's Example 5.1 produces a $[74,60,\geq10]$ code over $\mathbb{F}_{25}$, one longer than the codes in the earlier construction, giving a new best-known parameter entry.

Reading between the lines

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

A testable consequence not claimed by the paper: the same bottom-set compression should hold for any function field whose pure-gap set satisfies a translation-invariant floor-sum criterion, so the method could be adapted beyond Kummer extensions.
The arithmetic form of the criterion suggests a straightforward exhaustive check: recomputing the displayed 18-tuple bottom set in Example 5.1 from Lemma 3.2 would either confirm or refute the claimed record without any further theory.
Because Algorithm 3 reduces the task to $(m-1)^{s+1}$ floor-sum evaluations, it may make record-hunting over larger $m$ and more places computationally feasible, a direction the paper does not explore.
The equality of parameters with and without the infinite place in the tables hints that including $Q_\infty$ may not change the best achievable parameters for these families, although the paper does not claim this.

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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

Read Zhang–Zhao on pure gaps in Kummer extensions. The short version: the core theorem is solid, the record code is not yet supported, and Algorithm 3 has a typo that matters.

What's genuinely new: Theorem 3.1 — the bottom-set/tiling description of pure gaps at any number of totally ramified places in a Kummer extension with gcd(m,q)=1 — is a real step beyond the two-place results of [6] and the non-infinite descriptions of [8]. The one-inequality lifting criterion (Theorem 3.6) is a useful operational rule, and Propositions 4.2 and 4.3 give the first explicit infinite-place descriptions I know of. I spot-checked the key algebra: the cancellation in Theorem 3.1 works because the λ_i sum to zero; the two pure-gap tuples cited in Example 5.1 satisfy Lemma 3.2 at the boundary; the genus-7 arithmetic is consistent. So the method is not a mirage.

Now the soft spots. Algorithm 3 line 10 says Σ_{i=1}^s k_i ≤ −1−k, but Theorem 3.1 requires Σ_{i=0}^s k_i ≤ −1−k. As printed, the algorithm lets k_0 run free and returns a strict superset of G^0 whenever s≥1. That is a load-bearing bug in the pseudocode, even if the theorem itself is fine. Second, Example 5.1's 18-tuple set G^0(Q_1,Q_2) is asserted without derivation, code, or reproducibility check. The authors say 'Using Algorithm 3', but if they ran Algorithm 3 as printed, the set would be wrong; if they ran a corrected version, the paper doesn't say so. Since the [74,60,≥10] record depends on that set and on the whole coordinate-wise box being pure, the record claim currently hangs on an unverified computation. This is fixable — provide the computation or code — but it is not optional. Minor points: a line in the proof of Theorem 3.6 looks circular and is almost certainly a typo for '(a_0,...,a_{s-1},a)'; the 'efficient' claim is never quantified; and Lemma 3.2 is imported from [6] without re-derivation, which is fine as long as the reader knows where to look.

Who this is for: anyone working on pure gaps, Weierstrass semigroups for Kummer extensions, or multi-point AG codes. The paper deserves a serious referee, but the referee should be asked to check Algorithm 3 against Theorem 3.1 and to demand the Example 5.1 computation. After those fixes I'd be happy to see it in print.

Would I bring it to reading group? Yes — the gap between the theorem and the algorithm is instructive.

Referee Report

4 major / 5 minor

Summary. The paper studies Kummer extensions F=K(x,y) defined by y^m=∏(x−α_i)^{λ_i} with gcd(m,q)=1, and develops a method for describing all pure gaps at several totally ramified rational places. It introduces a finite "bottom set" of pure gaps and states (Theorem 3.1) that every pure gap is obtained from a bottom gap by adding multiples of m to the coordinates, with the total shift bounded by an explicit integer k_a. The paper gives three algorithms: one for the bottom gap set at one place, one for extending a bottom set by an additional place, and one for producing the full set of pure gaps. It then gives explicit closed-form descriptions in the equal-multiplicity case, including the infinite place, and applies the results to construct multi-point AG codes. The headline applied result is Example 5.1: a [74,60,≥10] differential AG code over F_25 from y^8=(x+1)^3(x^2+x+2)^7, claimed to be a new record against MinT's tables.

Significance. If the main theorem and algorithms are correct, the paper provides a unified algorithmic treatment of pure gaps at an arbitrary number of totally ramified places for arbitrary Kummer extensions, going beyond earlier work that mostly assumed equal multiplicities. The reduction to a finite bottom set is a useful structural idea, and Section 4's explicit formulas extend recent results by including the infinite place. The code constructions in Section 5 are concrete and potentially valuable; in particular, the claimed [74,60,≥10] code is a meaningful benchmark if the underlying pure-gap assertions are valid. My spot-checks confirm that the two tuples (2,9) and (10,1) used in Example 5.1 satisfy the pure-gap inequalities in Lemma 3.2, so the specific code parameters are credible. However, as printed, the central theorem and two of the algorithms contain sign/index errors, and the computation behind the record is not reproducible from the manuscript.

major comments (4)

[§3.1, Theorem 3.1] The definition of k_a has a sign error: the second sum is written with a minus sign, k_a = −1 − max_v(Σ_{i=0}^s floor((a_i−a_vσ_{j_v}λ_{j_i})/m) − Σ_{i=s+1}^r floor((−a_vσ_{j_v}λ_{j_i})/m)), but Lemma 3.2 and the proof of Theorem 3.1 require a plus sign before the second sum. With the printed minus sign, the bottom tuple (2,9) used in Example 5.1 gives max_v = 34, hence k_a = −35, so the claimed set G0(Q1,Q2) would be empty. The theorem is correct with '+Σ_{i=s+1}^r floor((−a_vσ_{j_v}λ_{j_i})/m)' inside the maximum; the sign must be corrected.
[§3.2, Algorithm 3, line 10] Algorithm 3 imposes Σ_{i=1}^s k_i ≤ −1−k, whereas Theorem 3.1 requires Σ_{i=0}^s k_i ≤ −1−k. As printed, k_0 is unbounded and the algorithm returns an infinite superset of the true pure-gap set for every s≥1. This invalidates the "find all pure gaps" claim and makes the 18-tuple computation in Example 5.1 irreproducible from the printed pseudocode. If the version used in Example 5.1 was corrected, the pseudocode must be amended to match it.
[§3.2, Algorithm 2 vs Proposition 3.5] Algorithm 2 adds (a0,…,a_{s−1},a) as soon as the single inequality on line 5 holds, but Proposition 3.5, which is cited as its justification, requires in addition that for every 1≤k≤a−1 at least one of inequality (2) or the non-congruence condition (3) holds. The else-break on lines 7–8 does not implement this check, and the same symbol a is used for the input tuple and the loop variable. As written, the algorithm can over-report the bottom set, and the computation of G0(Q1,Q2) in Example 5.1 is not certified by the proposition.
[§5, Example 5.1] The displayed 18-element set G0(Q1,Q2) is asserted without derivation, code, or a reproducible computation. I checked directly from Lemma 3.2 that the two tuples (2,9) and (10,1) satisfy the required inequalities, so the claimed parameters [74,60,≥10] are plausible; however, the completeness of the listed bottom set, and hence the claim that Algorithms 1–3 exactly compute G0 in this instance, is not certified by the manuscript. A verifiable computation, a short script, or a proof of set equality should be supplied before the record claim is treated as established.

minor comments (5)

[§3.2, proof of Theorem 3.6] In the case a≥2 the proof says "since (a0,…,a_{s−1},a+1)∈G0", but the relevant hypothesis is that (a0,…,a_{s−1},a)∈G0; the displayed argument otherwise goes through with the tuple (a0,…,a_{s−1},a). This appears to be a typo and should be corrected.
[§4, Proposition 4.5 and Corollary 4.9] Proposition 4.5 states "For each 1≤i≤m", which should be "1≤i≤r". In Corollary 4.9, the set S_{s+1} is defined as the set of permutations of {1,…,s}, but it should be the permutations of {0,…,s}.
[§3.2, Algorithm 2] The variable a is used both for the input tuple (a0,…,a_{s−1}) and for the inner loop variable ranging from 1 to m−1. This shadowing makes the pseudocode difficult to read and should be fixed by renaming one of the variables.
[Throughout] There are numerous typos and formatting errors, including "foucsed" (Section 1), "llustrate" (Section 5), "T able" in the table captions, "Propositons" in several places, "komw" in the proof of Proposition 4.10, and "greateat commom divisor" in Section 2. A careful proofreading pass is needed.
[§5, Example 5.1] The example should state explicitly that x^2+x+2 splits over F_25, so that the curve has the three finite totally ramified places with multiplicities 3, 7, and 7; this is needed to verify the stated genus g=7 and the place count N=76 from the data given.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain rests on an external characterization from [6] and standard function-field facts; no fitted quantity is renamed as a prediction.

full rationale

The central claim Theorem 3.1 is proved by deriving two inclusions from Lemma 3.2, which is itself an explicit restatement of the external criterion [6, Theorem 3.3], quoted in the paper as Theorem 2.7. That criterion is imported rather than established in this paper, and all later results explicitly inherit its hypotheses, so the paper does not assume its own conclusion. The bottom set in Definition 2.6 is exactly the pure-gap set intersected with the coordinate box [1,m-1]^{s+1}; Theorem 3.1's statement that the full pure-gap set is determined by the bottom set is therefore a substantive structural result, not a restatement of the definition. Propositions 3.3, 3.5 and 3.6 are proved using Lemma 3.2 and elementary floor-sum identities, with no parameter fitted to data and no target tuple inserted as a hypothesis. The applied record claim in Example 5.1 lists G0(Q1,Q2) without showing the computation, and Algorithm 3's printed summation bound appears incorrect, but an unshown or buggy computation is a correctness and reproducibility concern, not a circularity. There is no load-bearing self-citation, and the comparison to MinT's tables is external. Consequently, no circular step can be exhibited with the quoted reduction required by the analysis rules.

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

No free parameters are introduced: every quantity (floor sums, thresholds k_a, bounds t_{i,k}) is computed from m, r, and the λ_i, which are inputs of the curve definition; nothing is fitted to data or tuned to make code parameters come out. No new physical or mathematical entities are postulated: the 'bottom set' (Definition 2.6) is a defined subset of the existing pure-gap set, not an independent postulated object. The main external inputs are [6, Theorem 3.3] and standard Riemann-Roch/genus facts.

assumptions (4)

domain assumption Theorem 2.7 ([6, Theorem 3.3]): (a_0,...,a_s) is a pure gap iff for each t modulo m exactly one of two floor-sum conditions holds.
The paper's Lemma 3.2 and Theorem 3.1 are derived from this external criterion, quoted in Section 2 without proof. Its validity for arbitrary λ_i is assumed, and it is the operational definition of pure gaps throughout the paper.
standard math Genus formula g = (m(r-1)+2-Σgcd(m,λ_i))/2 for y^m = ∏(x-α_i)^{λ_i}, from [19, Proposition 3.7.3].
Used in Section 2 and in all code parameter computations in Section 5 (e.g., g = 7 and the deg G bounds). It is a standard textbook result whose conditions (gcd(m,q)=1, distinct α_i) are stated by the authors.
domain assumption For gcd(m,λ_i)=1 the place over x = α_i is rational and totally ramified, with pole divisor identities (x-α_i)^{-k} = km/gcd(m,λ_0)Σ... - kmQ_i (Section 2).
The bottom-set definition and the claim km ∉ G(Q_i) rely on these identities. The theory is silent for places with gcd(m,λ_i)>1, which restricts the scope to totally ramified places.
standard math Riemann-Roch characterization of gaps and pure gaps (ℓ(Σa_iQ_i) = ℓ(Σa_iQ_i - Q_j) for all j) and the minimum-distance bound Theorem 2.2 from [3, Theorem 3.4].
These connect the algebraic definition of pure gaps to code parameters in Section 5 and are used as black boxes throughout the constructions.

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Cite this review

Pith. "Pith review of Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions." pith.science (2026). https://pith.science/paper/BWKFHY4W

@misc{pith2026250523274,
  author       = {Pith},
  title        = {Pith review of: Pure Gaps at Many Places and Multi-point AG Codes from Arbitrary Kummer Extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWKFHY4W}},
  note         = {Machine review of arXiv:2505.23274}
}

abstract

For a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\a_i)^{\lambda_i}$ over an algebraic extension $K$ of a finite field $\fq$, where $\la_i\in \Z\backslash\{0\}$ for $1\leq i\leq r$, $\gcd(m,q) = 1$, and $\a_1,\cdots,\a_r\in K$ are pairwise distinct elements, we propose a simple and efficient method to find all pure gaps at many totally ramified places. We introduce a bottom set of pure gaps and indicate that the set of pure gaps is completely determined by the bottom set. Furthermore, we demonstrate that a pure gap can be deduced from a known pure gap by easily verifying only one inequality. Then, in the case where $\lambda_1 = \lambda_2 = \cdots = \lambda_r$, we fully determine an explicit description of the set of pure gaps at many totally ramified places, This includes the scenario in which the set of these places contains the infinite place. Finally, we apply these results to construct multi-point algebraic geometry codes with good parameters. As one of the examples, a presented code with parameters $[74, 60, \geq 10]$ over $\mathbb{F}_{25}$ yields a new record.

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