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Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions

This paper establishes an arithmetic characterization of non-special divisors on Kummer extensions to explicitly construct effective divisors of degrees $g$ and $g-1$ , thereby developing a general framework for generating linear complementary pairs (LCPs) and linear complementary dual (LCD) algebraic geometry codes with determined security parameters, including specific applications to the GK and Hermitian curves.

Original authors: Huachao Zhang, Chang-An Zhao

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Huachao Zhang, Chang-An Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are building a high-security vault system for digital information. To make this vault unbreakable, you need two things: a very strong lock (to stop thieves from picking it) and a backup key that is completely different from the lock but fits the same door (to stop hackers from copying the key). In the world of mathematics and coding theory, these are called LCD codes (the lock) and LCPs (the lock and backup key pair).

This paper is like a master blueprint for building these "locks and keys" using a specific type of mathematical landscape called Kummer extensions. Here is a simple breakdown of what the authors did, using everyday analogies.

1. The Landscape: Kummer Extensions

Think of a Kummer extension as a special kind of multi-layered map. Usually, maps are flat, but this one is like a spiral staircase or a multi-story parking garage built over a simple road (a function field).

The road is defined by an equation like $y^m = f(x)$ .
The "floors" of the garage are the different values $y$ can take for a given $x$ .
The authors are interested in specific spots on this map called places (like specific parking spots or street corners). Some spots are "totally ramified," meaning all the floors merge into one single spot (like a funnel). Others are "non-totally ramified," where the floors stay distinct.

2. The Problem: Finding the "Non-Special" Keys

To build the secure codes, the mathematicians need to find specific collections of spots on this map called divisors.

Special Divisors: These are like "bad keys" or "broken locks." They don't work well for the security system because they have too much redundancy or don't fit the mathematical rules required for security.
Non-Special Divisors: These are the "perfect keys." They are rare and precise. The paper focuses on finding these perfect keys that have a specific size (degree) and can be placed in tricky spots (including those where the floors don't merge).

The Breakthrough:
Previous researchers could only find these perfect keys in the "funnel" spots (totally ramified places). The authors of this paper invented a new arithmetic recipe (Theorem 17) that allows them to find these perfect keys even in the complex spots where the floors don't merge. It's like discovering a way to find a perfect key in a crowded, messy parking lot, not just in the empty, organized garage.

3. The Construction: Building the Vaults

Once they found the recipe for the perfect keys, they used them to build two types of security systems:

A. Linear Complementary Pairs (LCPs)

Imagine you have a main vault door (Code A) and a backup door (Code B).

Together, they cover every possible way to enter the building (they sum up to the whole space).
They don't overlap in a way that creates a weak spot.
The "security parameter" is determined by how hard it is to break either door. The authors showed how to calculate exactly how strong these doors are based on the perfect keys they found.

B. Linear Complementary Dual (LCD) Codes

This is a single, super-strong vault door where the lock and the key are mathematically "opposites" of each other. If you try to copy the key, it doesn't work because the lock is designed to reject its own reflection. The authors showed how to build these using their new recipe.

4. The Specific Examples: The GK Curve and Hermitian Quotients

The authors didn't just write a theory; they tested it on famous mathematical shapes:

The GK Curve: Think of this as a very complex, high-security fortress. The authors successfully built a set of perfect keys specifically for this fortress, proving their recipe works even on the most complicated maps.
Hermitian Curve Quotients: These are slightly simpler but still very secure shapes. The authors built families of locks and keys here too, showing that their method is flexible.

5. The "Pure Gaps" Trick

To find some of these perfect keys, the authors used a concept called pure gaps.

Imagine a staircase where some steps are missing. A "gap" is a missing step.
A "pure gap" is a missing step that is so missing that you can't even pretend it's there to help you climb.
The authors realized that if they know exactly where these "pure missing steps" are, they can mathematically guarantee the existence of a perfect key (a non-special divisor) right next to them.

Summary

In short, this paper is a construction manual.

It gives a new formula to find the rare, perfect mathematical objects (non-special divisors) needed for secure coding.
It proves this formula works even in the most complicated, messy parts of the mathematical landscape.
It uses these objects to build new, highly secure digital codes (LCPs and LCDs) on famous mathematical shapes.
It provides concrete examples showing exactly how to build these codes, giving specific numbers for how strong the security is.

The authors are essentially saying: "We found a new way to find the perfect ingredients, and here is the exact recipe to bake the most secure digital cakes possible using those ingredients."

Technical Summary: Non-special Divisors, LCPs of Codes, and LCD Codes on Kummer Extensions

Problem Statement
Linear Complementary Pairs (LCPs) of codes and Linear Complementary Dual (LCD) codes are critical in coding theory and cryptography, particularly for protecting against side-channel attacks (SCA) and fault injection attacks (FIA). The construction of such codes on function fields relies heavily on the existence and characterization of non-special divisors of degrees $g$ and $g-1$ , where $g$ is the genus of the function field. While previous works have established constructions for specific Kummer extensions (e.g., those defined by $y^m = \prod (x-\alpha_i)$ with totally ramified places), there remains a need for a generalized arithmetic characterization of non-special divisors that can accommodate arbitrary degrees and supports containing non-totally ramified or non-rational places. Furthermore, explicit constructions of these divisors on complex curves, such as the Giulietti-Korchmáros (GK) curve, and the development of a unified framework for generating LCPs and LCD codes on Kummer extensions are required.

Methodology
The authors employ the theory of algebraic geometry (AG) codes, Kummer extensions, and Riemann-Roch spaces. The methodology proceeds in three main stages:

Arithmetic Characterization: The paper investigates Kummer extensions $F = \mathbb{F}_q(x, y)/\mathbb{F}_q(x)$ defined by $y^m = f(x)$ , where $f(x) \in \mathbb{F}_q(x) \setminus \mathbb{F}_q$ . By unifying the extensions of places under Galois group actions and utilizing dimension formulas derived from constant field extensions, the authors derive a necessary and sufficient arithmetic condition for a divisor $D = \sum a_i D_i$ to be non-special. This condition involves floor functions and the degrees of places, generalizing previous results that were restricted to totally ramified places.
Construction via Canonical Divisors and Pure Gaps: The authors utilize the relationship between a non-special divisor of degree $g-1$ and a canonical divisor $W$ to generate new non-special divisors. Additionally, they employ the theory of pure gaps (elements in the complement of the Weierstrass semigroup) to characterize effective non-special divisors of degree $g$ .
Framework for Code Construction: A general framework is developed to construct LCPs of AG codes $(C_L(D, G), C_L(D, H))$ and LCD codes. This framework relies on selecting divisors $G$ and $H$ such that their greatest common divisor and least common multiple satisfy specific non-special conditions. The security parameters of the resulting LCPs are shown to be determined by the minimum distance of one of the constituent codes.

Key Contributions and Results

Generalized Characterization (Theorem 17): The paper presents a generalized arithmetic characterization for non-special divisors on Kummer extensions. Unlike prior work, this characterization allows for divisors of arbitrary degree and supports that may include non-totally ramified places and non-rational places.
Explicit Constructions on the GK Curve: Using the derived characterization, the authors explicitly construct a non-special divisor of degree $g-1$ on the function field of the GK curve (Proposition 24). This addresses a gap where previous methods failed to find such divisors with support in the standard set of rational places.
Families of Effective Non-Special Divisors: For Kummer extensions defined by $y^m = \prod (x-\alpha_i)^\lambda$ with identical multiplicities, the authors construct three distinct families of effective non-special divisors of degree $g$ (Propositions 25, 26, and 27). These families cover cases where $m > r$ , $m < r$ , and general $m, r$ , and are proven to be inequivalent.
General Framework for LCPs and LCDs: Theorem 29 and Corollary 30 provide a systematic method for constructing LCPs of AG codes on Kummer extensions. The framework demonstrates that the security parameter of the LCP is given by the minimum distance of one of the codes. Furthermore, it establishes sufficient conditions for constructing LCD AG codes, specifically utilizing linearized polynomials (Proposition 37).
Concrete Examples: The theoretical results are illustrated with explicit examples, including:
- LCPs of AG codes on the GK curve (Proposition 31).
- LCPs and LCD codes on quotients of the Hermitian curve (Corollary 38, Example 39).
- Specific parameter sets for codes derived from linearized polynomials.

Significance and Claims
The paper claims to extend the state of the art in the construction of AG codes by removing the restriction that non-special divisors must have support consisting solely of totally ramified places. By providing a unified arithmetic criterion, the work enables the identification of non-special divisors in broader contexts. The explicit construction of non-special divisors on the GK curve is highlighted as a significant advancement, as previous attempts using different techniques failed to locate such divisors with the required properties.

The development of a general framework for LCPs and LCD codes allows for the systematic generation of code families with known security parameters. The authors note that while Theorem 17 characterizes these divisors, the general existence problem for such divisors remains an open question. Additionally, the paper suggests that future work could explore varying the choice of the index set $I$ and the Weil differential $\omega$ to generate further families of codes, but currently, the presented results focus on specific, verified constructions. The work is grounded in the existing literature on AG codes, LCPs, and LCD codes, building upon the foundational characterizations provided by earlier researchers.