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arxiv: 2605.29439 · v2 · pith:YW7GLOBBnew · submitted 2026-05-28 · 💻 cs.IT · math.IT

On the Maximal Length of MDS Elliptic Codes

Haojie Chen , Chuangqiang Hu , Junjie Huang , Chang-An Zhao This is my paper

Pith reviewed 2026-06-29 05:54 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords MDS codeselliptic curvesalgebraic geometry codesmaximal lengthfinite fieldscoding theoryRiemann-Roch
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The pith

MDS elliptic codes over finite fields attain their maximal length of (q+1 + floor(2 sqrt(q)))/2 when even and (q + floor(2 sqrt(q)))/2 when odd.

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

The paper determines the exact maximal length of MDS codes constructed from elliptic curves over any finite field F_q. It shows that previous upper bounds are achieved in all cases by suitable constructions, resolving the question for even dimensions and non-square fields. This completes the determination of how long such optimal codes can be. Sympathetic readers care because it fixes the best error-correcting performance obtainable from elliptic curves in this setting.

Core claim

If the support of the divisor G consists of F_q-rational points, the bound on MEC(k,q) decreases to (q+1)/2 + sqrt(q) -1 for even k. Without this restriction, MDS codes attaining (q+1)/2 + sqrt(q) exist for even k. In general, MEC(k,q) equals (q+1 + floor(2 sqrt(q)))/2 when q+1 + floor(2 sqrt(q)) is even, and (q + floor(2 sqrt(q)))/2 when odd.

What carries the argument

The Riemann-Roch theorem applied to the divisor class group of the elliptic curve, distinguishing cases where the support points are all defined over F_q versus over algebraic closures.

If this is right

  • For even k with only rational points in the support, the maximal length is reduced by one.
  • Constructions using general points achieve the higher bound for even k.
  • The result holds in characteristic two and for non-square q.
  • All open cases for the tightness of the bound are resolved.

Where Pith is reading between the lines

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

  • The ability to use points over quadratic extensions allows increasing the code length by one when k is even.
  • This distinction between rational and non-rational points may have analogues in MDS codes from other algebraic curves.
  • The formula provides a concrete way to compute the best possible length for given q without case-by-case analysis.

Load-bearing premise

The proofs rely on the divisor class group and Riemann-Roch theorem giving exact control over the minimum distance of the resulting codes.

What would settle it

An explicit MDS elliptic code whose length exceeds the value given by the formula for some q and k, or a demonstration that the stated length cannot be achieved in a particular case.

read the original abstract

The determination of the maximal length of maximum distance separable (MDS) codes arising from elliptic curves is a central problem in coding theory. For an elliptic curve $E$ over $\mathbb{F}_q$, let $\operatorname{MEC}(k,q)$ denote the maximal length of a $q$-ary MDS elliptic code of dimension $k$. It was recently shown that $\operatorname{MEC}(k,q)\le\frac{q+1}{2}+\sqrt{q}$ for $q\ge289$ and $3\le k\le(q+1-2\sqrt{q})/10$, with equality for odd $k$ when $q$ is an odd square. This paper investigates the remaining open cases, namely even dimension $k$, non-square $q$ and fields of characteristic $2$, and provides a complete resolution of the tightness question for the two natural parity regimes of $q+1+\lfloor 2\sqrt{q}\rfloor$. We prove that if the support of $G$ (used to define the code) consists of $\mathbb{F}_q$-rational points, the bound decreases to $\frac{q+1}{2}+\sqrt{q}-1$ for even $k$. Without this restriction, we construct MDS codes attaining $\frac{q+1}{2}+\sqrt{q}$ for even $k$. More generally, we establish $\operatorname{MEC}(k,q)=\frac{q+1+\lfloor2\sqrt{q}\rfloor}{2}$ when $q+1+\lfloor2\sqrt{q}\rfloor$ is even, and $\operatorname{MEC}(k,q)=\frac{q+\lfloor2\sqrt{q}\rfloor}{2}$ when it is odd.

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

0 major / 3 minor

Summary. The manuscript resolves the remaining open cases for the maximal length MEC(k,q) of q-ary MDS elliptic codes of dimension k. It proves an upper bound of (q+1 + floor(2 sqrt(q)))/2 or (q + floor(2 sqrt(q)))/2 depending on the parity of q+1 + floor(2 sqrt(q)), using the Riemann-Roch theorem on elliptic curves, and provides explicit constructions attaining these values. The bound tightens by 1 when the support of the divisor G consists solely of F_q-rational points and k is even; without this restriction the full value is achieved even for even k. The results cover even k, non-square q, and characteristic 2, completing the determination of MEC(k,q) in the regime q >= 289 and 3 <= k <= (q+1-2 sqrt(q))/10.

Significance. If the upper-bound arguments and constructions hold, the paper supplies a complete, tight determination of MEC(k,q) for the previously unresolved parity and field cases, extending the earlier result that achieved equality only for odd k when q is an odd square. The explicit distinction between rational-point supports and general algebraic-closure supports, together with matching constructions, strengthens the theory of algebraic-geometry codes from elliptic curves and furnishes falsifiable predictions for code lengths.

minor comments (3)
  1. The abstract states the main equalities but does not explicitly restate the range 3 <= k <= (q+1-2 sqrt(q))/10 under which the new results apply; adding this clarification in the introduction would help readers connect the new theorems to the prior bound.
  2. The notation floor(2 sqrt(q)) is used without defining the floor function on first appearance; a brief parenthetical or footnote would improve readability for readers outside algebraic coding theory.
  3. The abstract refers to 'the two natural parity regimes of q+1 + floor(2 sqrt(q))' but does not indicate whether the constructions are uniform across all q satisfying the inequality or require case-by-case verification; a short remark on uniformity would be helpful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript, including the accurate summary of our results on MEC(k,q) and the recommendation for minor revision. The referee's assessment aligns with the scope of the work, which completes the determination of the maximal length for the remaining cases of even k, non-square q, and characteristic 2.

Circularity Check

0 steps flagged

No significant circularity; upper bounds from Riemann-Roch and explicit constructions are independent

full rationale

The derivation combines the Riemann-Roch theorem and divisor class group structure on elliptic curves (standard AG code theory) for upper bounds with explicit point selections for constructions achieving the stated MEC(k,q) values. The parity distinction and rational-point support condition are invoked directly in the proofs without reducing any claimed length to a fitted parameter or self-referential quantity. The 'recently shown' bound is cited only for context on prior cases; the new results for even k, non-square q, and char 2 are derived from first principles in the paper. No self-citation is load-bearing for the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard algebraic-geometry machinery for elliptic curves over finite fields (Riemann-Roch, divisor class group, Weierstrass form) and the definition of algebraic-geometry codes; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • standard math Riemann-Roch theorem and properties of the Picard group of an elliptic curve over a finite field
    Invoked implicitly when defining the code from a divisor G and when bounding the minimum distance.
  • standard math Existence of elliptic curves with prescribed numbers of rational points (Hasse bound)
    Used to obtain the floor(2 sqrt(q)) term appearing in the final formulas.

pith-pipeline@v0.9.1-grok · 5838 in / 1502 out tokens · 28493 ms · 2026-06-29T05:54:13.075825+00:00 · methodology

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