pith. sign in

arxiv: 1906.10620 · v1 · pith:C3RIM5BNnew · submitted 2019-06-25 · 💻 cs.IT · math.AG· math.IT

Isometry-Dual Flags of AG Codes

Maria Bras-Amor\'os , Iwan Duursma , Euijin Hong This is my paper

Pith reviewed 2026-05-25 15:57 UTC · model grok-4.3

classification 💻 cs.IT math.AGmath.IT
keywords algebraic geometry codesisometry-dual flagsone-point AG codesWeierstrass semigroupAG codesself-dual codesalgebraic curvesfinite geometry
0
0 comments X

The pith

A flag of one-point AG codes is isometry-dual if and only if its final code uses functions with pole orders at most n + 2g - 1, for n at least 2g + 2.

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

The paper establishes a complete characterization for when a flag of one-point algebraic geometry codes has the isometry-dual property. For any subset of n points with n at least twice the genus plus two, this property holds exactly when the largest code uses functions whose poles at Q have order no higher than n plus twice the genus minus one. This extends a previous result limited to larger n and shows the bound is sharp by providing counterexamples for smaller n. The result also includes a necessary condition for when puncturing preserves the property, based on the Weierstrass semigroup.

Core claim

For a curve of genus g, a flag of one-point AG codes defined with a subset of n ≥ 2g+2 rational points is isometry-dual if and only if the last code Cn in the flag is defined with functions of pole order at most n+2g-1. This characterization is extended from the case n > 2g+2 using a different approach based on the Weierstrass semigroup. Examples show that for n=2g+1 the property can hold with a smaller pole order bound of n+2g-2. A necessary condition in terms of maximum sparse ideals of the Weierstrass semigroup is given for punctured flags to inherit the isometry-dual property.

What carries the argument

The isometry-dual property of a flag of one-point AG codes, which requires the flag to coincide with its dual up to isometry and is characterized by the maximum pole order in the space generating the final code Cn.

If this is right

  • For n ≥ 2g+2 the isometry-dual property of the flag is completely determined by the pole-order bound on Cn.
  • The characterization does not extend unchanged to n = 2g+1, as shown by explicit examples where the bound can be lowered by one.
  • A punctured flag inherits the isometry-dual property from the original only when the Weierstrass semigroup satisfies the stated condition on its maximum sparse ideals.

Where Pith is reading between the lines

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

  • The pole-order criterion may simplify the search for isometry-dual flags on other curves with known Weierstrass semigroups.
  • The semigroup condition on puncturing could connect the result to constructions that delete coordinates while preserving duality properties.
  • Verification on the listed curves (Hermitian, Suzuki, Ree, Klein) might produce explicit infinite families satisfying the new bound.

Load-bearing premise

The isometry-dual property of the maximal flag on the given curves follows from the Weierstrass semigroup structure at Q.

What would settle it

A flag with n ≥ 2g+2 where Cn is generated by functions of pole order at most n+2g-1 but the flag and its dual are not isometric, or the converse case.

Figures

Figures reproduced from arXiv: 1906.10620 by Euijin Hong, Iwan Duursma, Maria Bras-Amor\'os.

Figure 1
Figure 1. Figure 1: Dimension-degree pairs (n, m) admitted by Theorem 10 Or equivalently, # of nongaps in [1, a] ≤ a 2 ≤ # of gaps in [1, a] # of gaps in [a + 1, 2g] ≤ 2g − a 2 ≤ # of nongaps in [a + 1, 2g]. Proof. The first inequality follows from the Clifford’s Theorem applied to the divisor aQ. The second one is its complement to the fact that there are exactly g Weierstrass gaps nongaps in [1, 2g]. Theorem 10 (Riemann-Roc… view at source ↗
Figure 2
Figure 2. Figure 2: Dimension-degree pairs (n, m) admitted by Proposition 6 [m−2g, 2g] such that u+v = m. So, suppose that there are equal number of Weierstrass gaps and nongaps in the interval [m − 2g, 2g]. Then again by Lemma 9, the number m − 2g − 1 is a nongap. Then (u, v) = (m − 2g − 1, 2g + 1) satisfies the condition. Proof of Proposition 6. In the interval [0, m], there are at most 2g geometric nongaps. So, n ≥ m + 1 −… view at source ↗
read the original abstract

Consider a complete flag $\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$ of one-point AG codes of length $n$ over the finite field $\mathbb{F}$. The codes are defined by evaluating functions with poles at a given point $Q$ in points $P_1,\dots,P_n$ distinct from $Q$. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point $Q$, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In [3] it is shown, for a curve of genus $g$, that a flag of one-point AG codes defined with a subset of $n > 2g+2$ rational points is isometry-dual if and only if the last code $C_n$ in the flag is defined with functions of pole order at most $n+2g-1$. Using a different approach, we extend this characterization to all subsets of size $n \geq 2g+2$. Moreover we show that this is best possible by giving examples of isometry-dual flags with $n=2g+1$ such that $C_n$ is generated by functions of pole order at most $n+2g-2$. We also prove a necessary condition, formulated in terms of maximum sparse ideals of the Weierstrass semigroup of $Q$, under which a flag of punctured one-point AG codes inherits the isometry-dual property from the original unpunctured flag.

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 / 2 minor

Summary. The paper extends the if-and-only-if characterization of isometry-dual flags of one-point AG codes from the case n > 2g+2 (previously shown in [3]) to all subsets with n ≥ 2g+2. A flag is isometry-dual precisely when the terminal code Cn is generated by functions of pole order at most n+2g−1. The extension uses Weierstrass semigroups at Q together with maximum sparse ideals; the authors also exhibit sharpness via explicit n=2g+1 examples where the pole-order bound can be relaxed to n+2g−2, and they derive a necessary condition (in terms of maximum sparse ideals) for a punctured flag to inherit the isometry-dual property from its unpunctured parent.

Significance. If the central claims hold, the work supplies a uniform, semigroup-based criterion that applies to all curves admitting a rational point Q with the stated Weierstrass properties and covers both maximal and proper subsets of rational points. The new approach via sparse ideals is independent of the earlier method in [3] and yields concrete sharpness examples on the projective line, Hermitian, Suzuki, Ree, and Klein curves. These results tighten the known boundary between self-dual and non-self-dual flags and provide a necessary condition for puncturing that may be reusable in other AG-code constructions.

major comments (1)
  1. [Theorem 3.4 / proof of the only-if direction] The extension of the iff statement to the boundary n=2g+2 is load-bearing for the main theorem. The argument in the proof that maximum sparse ideals remain maximal when the non-gap count becomes an equality (the case n=2g+2) must be checked against the explicit dimension formulas; if any subset-dependent extra condition appears at equality, the claimed iff would require an additional hypothesis not stated in the theorem.
minor comments (2)
  1. [Section 2] Notation for the Weierstrass semigroup and the associated sparse ideals is introduced without a consolidated table of symbols; a short notation table would improve readability.
  2. [Section 5] The sharpness examples for n=2g+1 are stated for specific curves but the pole-order generators are not listed explicitly; adding one concrete generator list per curve would make the sharpness claim immediately verifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the boundary case n=2g+2 in Theorem 3.4. The comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [Theorem 3.4 / proof of the only-if direction] The extension of the iff statement to the boundary n=2g+2 is load-bearing for the main theorem. The argument in the proof that maximum sparse ideals remain maximal when the non-gap count becomes an equality (the case n=2g+2) must be checked against the explicit dimension formulas; if any subset-dependent extra condition appears at equality, the claimed iff would require an additional hypothesis not stated in the theorem.

    Authors: The only-if direction of Theorem 3.4 proceeds by showing that if the flag is isometry-dual then the terminal code Cn must be generated by functions of pole order at most n+2g-1. At the boundary n=2g+2 the proof invokes the explicit Riemann-Roch dimension formulas together with the definition of a maximum sparse ideal in the Weierstrass semigroup at Q. These formulas yield that the codimension of the ideal is exactly the number of non-gaps up to n+2g-1, and maximality of the sparse ideal follows directly from the semigroup addition law without reference to the particular choice of the n points. Consequently no subset-dependent extra condition arises at equality; the same algebraic criterion that works for n>2g+2 continues to hold verbatim at n=2g+2. The argument is therefore uniform and the stated iff characterization requires no additional hypothesis. revision: no

Circularity Check

0 steps flagged

No significant circularity; extension uses independent approach

full rationale

The paper cites [3] only for the strict n > 2g+2 base case and explicitly states it employs a different approach (Weierstrass semigroups at Q and maximum sparse ideals) to prove the extension to n ≥ 2g+2. The iff characterization, necessary conditions for punctured flags, and sharpness examples at n=2g+1 are derived directly from these tools rather than by fitting parameters, redefining inputs, or reducing to the prior result by construction. The derivation chain is therefore self-contained and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions of algebraic-geometry coding theory and the structure of Weierstrass semigroups; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption One-point AG codes and their duals are defined by evaluation of functions with prescribed poles at a single point Q on a curve of genus g.
    This is the foundational setup for the flags considered throughout the abstract.
  • domain assumption The Weierstrass semigroup at Q encodes the possible pole orders that determine the dimensions and duality properties of the codes.
    Invoked when stating the pole-order threshold and the necessary condition for punctured flags.

pith-pipeline@v0.9.0 · 5887 in / 1628 out tokens · 41144 ms · 2026-05-25T15:57:34.418460+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Representation of numerical semigroups by D yck paths

    Maria Bras-Amor\' o s and Anna de Mier. Representation of numerical semigroups by D yck paths. Semigroup Forum , 75(3):677--682, 2007

    work page 2007

  2. [2]

    New lower bounds on the generalized H amming weights of AG codes

    Maria Bras-Amor\' o s, Kwankyu Lee, and Albert Vico-Oton. New lower bounds on the generalized H amming weights of AG codes. IEEE Trans. Inform. Theory , 60(10):5930--5937, 2014

    work page 2014

  3. [3]

    On the order bounds for one-point AG codes

    Olav Geil, Carlos Munuera, Diego Ruano, and Fernando Torres. On the order bounds for one-point AG codes. Adv. Math. Commun. , 5(3):489--504, 2011

    work page 2011

  4. [4]

    F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Mathematical Library, Vol. 16

    work page 1977

  5. [5]

    Equality of geometric G oppa codes and equivalence of divisors

    Carlos Munuera and Ruud Pellikaan. Equality of geometric G oppa codes and equivalence of divisors. J. Pure Appl. Algebra , 90(3):229--252, 1993

    work page 1993

  6. [6]

    New lower bounds on the generalized H amming weights of AG codes

    Maria Bras-Amor\'os, Kwankyu Lee, and Albert Vico-Oton. New lower bounds on the generalized H amming weights of AG codes. IEEE Trans. Inform. Theory , 60(10):5930--5937, 2014

    work page 2014

  7. [7]

    Equality of geometric Goppa codes and equivalence of divisors

    Carlos Munuera, and Ruud Pellikaan. Equality of geometric Goppa codes and equivalence of divisors. Journal of Pure and Applied Algebra , 90(3), 229--252, 1993

    work page 1993

  8. [8]

    The theory of error-correcting codes

    MacWilliams and Sloane. The theory of error-correcting codes. North-Holland Publishing Co., Amsterdam-New York-Oxford , 16, 1977

    work page 1977

  9. [9]

    Algebraic function fields and codes

    Henning Stichtenoth. Algebraic function fields and codes. Springer , 254, 2009

    work page 2009