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arxiv: 2604.25354 · v1 · submitted 2026-04-28 · 💻 cs.IT · math.IT

On the Minimum Distances of Some Families of Goppa Codes and BCH Codes

Yaqi Chen , Hao Chen , Cunsheng Ding , Huimin Lao This is my paper

Pith reviewed 2026-05-07 15:00 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords Goppa codesBCH codesminimum distancedesigned distancefinite fieldsalternant codeserror-correcting codesalgebraic coding theory
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The pith

A necessary and sufficient criterion determines exactly when Goppa codes reach designed distance t+1 and fixes the distances for multiple families including primitive BCH codes.

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

The paper develops a necessary and sufficient criterion that tells when a Goppa code achieves its designed minimum distance of t+1, where t is the degree of the Goppa polynomial. The authors apply the criterion to compute exact minimum distances for several classes of q-ary Goppa codes, prove tightness for wild Goppa codes, and extend a specific family to odd prime power fields. Specializing to monomial polynomials G(x) = x^t, which correspond to primitive BCH codes, produces infinite families where actual distance equals designed distance, with an improved divisibility condition for one family. Exact distances matter because they fix the error-correcting capability of these codes in applications such as communications and code-based cryptography.

Core claim

We provide a necessary and sufficient criterion for a Goppa code to attain its designed distance δ=t+1, where t is the degree of the Goppa polynomial. As applications, we determine the minimum distances of several classes of q-ary Goppa codes. In particular, we prove the tightness of the improved lower bound for a class of wild Goppa codes, and extend the family with G(x)=x^t+A from the binary case to arbitrary odd prime powers. We then specialize the criterion to the monomial case G(x)=x^t, which is equivalent to primitive BCH codes. This leads to several infinite families of primitive BCH codes with d=δ, including the binary codes C_{(2,2^m-1,9,1)} and C_{(2,2^m-1,15,1)}, the family C_{(p,

What carries the argument

The necessary and sufficient criterion for a Goppa code to attain designed distance t+1 based on properties of its Goppa polynomial of degree t.

Load-bearing premise

The Goppa code is built in the standard manner from a polynomial of exact degree t over the finite field without further constraints that would change the distance.

What would settle it

A Goppa code satisfying the criterion but possessing a nonzero codeword of weight at most t, or a code failing the criterion yet having minimum distance exactly t+1, would show the criterion is not necessary and sufficient.

read the original abstract

Goppa codes form an important class of alternant codes with wide applications in algebraic coding theory and code-based cryptography. Determining the true minimum distance of a Goppa code is a difficult problem. In this paper, we provide a necessary and sufficient criterion for a Goppa code to attain its designed distance $\delta=t+1$, where $t$ is the degree of the Goppa polynomial. As applications, we determine the minimum distances of several classes of $q$-ary Goppa codes. In particular, we prove the tightness of the improved lower bound for a class of wild Goppa codes, and extend the family with $G(x)=x^t+A$ from the binary case to arbitrary odd prime powers. We then specialize the criterion to the monomial case $G(x)=x^t$, which is equivalent to primitive BCH codes. This leads to several infinite families of primitive BCH codes with $d=\delta$, including the binary codes $\mathbf{C}_{(2,2^m-1,9,1)}$ and $\mathbf{C}_{(2,2^m-1,15,1)}$, the family $\mathbf{C}_{(p,p^p-1,2p+2,1)}$ with an odd prime $p$ and the family $\mathbf{C}_{(q,q^m-1,r\frac{q^m-1}{q-1}+1,1)}$ with $r\mid q-1$. In particular, we prove that the primitive BCH code $\mathbf{C}_{(q,q^m-1,q^t+1,1)}$ has minimum distance $q^t+1$ under the condition $t\mid m$, improving the previously known condition $pt\mid m$.

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 / 2 minor

Summary. The paper provides a necessary and sufficient criterion for a Goppa code to attain its designed distance δ = t + 1, where t is the degree of the Goppa polynomial. It applies this criterion to determine the exact minimum distances of several classes of q-ary Goppa codes, including a class of wild Goppa codes (proving tightness of an improved lower bound), the monomial case G(x) = x^t (equivalent to primitive BCH codes), and infinite families such as binary C_{(2,2^m-1,9,1)} and C_{(2,2^m-1,15,1)}, the family C_{(p,p^p-1,2p+2,1)} for odd prime p, the family C_{(q,q^m-1,r(q^m-1)/(q-1)+1,1)} with r | q-1, and the family C_{(q,q^m-1,q^t+1,1)} under the improved condition t | m.

Significance. If the criterion and its applications hold, the work strengthens results on the true minimum distances of Goppa and BCH codes, which are central to algebraic coding theory and code-based cryptography. It supplies exact distances for multiple infinite families and improves the divisibility condition for one BCH family from pt | m to t | m, offering concrete, falsifiable statements that can be checked against known tables or small-parameter computations.

minor comments (2)
  1. [§1] §1 (Introduction): the statement that the criterion is 'necessary and sufficient' for attaining δ = t + 1 should be accompanied by an explicit reference to the precise definition of the Goppa code (parity-check matrix or locator polynomial) used in the derivation, to avoid ambiguity when the Goppa polynomial is not square-free.
  2. [Applications to BCH codes] The proof of the improved condition t | m for the family C_{(q,q^m-1,q^t+1,1)} is presented as a direct specialization; a short remark on whether the same argument yields the result for composite t not dividing m would clarify the sharpness of the new bound.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately reflects the main results: the necessary and sufficient criterion for Goppa codes to achieve minimum distance equal to the designed distance, along with its applications to wild Goppa codes, monomial Goppa codes equivalent to primitive BCH codes, and several infinite families with exact distances.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from standard Goppa code definitions

full rationale

The central result is a necessary and sufficient criterion for a Goppa code to attain designed distance δ = t + 1, derived directly from the standard definition of Goppa codes via the Goppa polynomial of degree t and the properties of the underlying finite field. Applications consist of specializing this criterion to concrete polynomial forms (e.g., G(x) = x^t, wild Goppa codes, monomial cases equivalent to primitive BCH codes) and verifying the distance condition under stated divisibility constraints such as t | m. No parameters are fitted to data and then relabeled as predictions, no self-citations supply load-bearing uniqueness theorems, and no ansatz or renaming reduces the claimed result to its own inputs by construction. The derivation chain remains independent of the target families.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are introduced in the abstract; the work relies on established mathematical structures in coding theory.

axioms (1)
  • standard math Standard properties of finite fields and polynomial rings over them
    The paper operates within the framework of algebraic coding theory over finite fields.

pith-pipeline@v0.9.0 · 5614 in / 1139 out tokens · 55132 ms · 2026-05-07T15:00:42.953871+00:00 · methodology

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