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Equivalence of Families of Polycyclic Codes over Finite Fields

This paper introduces an $n$ -equivalence relation to classify families of polycyclic codes over finite fields, deriving conditions for their equivalence and proving that $\ell$ -trinomial codes with $\gcd(n, n-\ell)=1$ are equivalent to a specific standard form.

Original authors: Hassan Ou-azzou, Anna-Lena Horlemann

Published 2026-05-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hassan Ou-azzou, Anna-Lena Horlemann

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 a master architect trying to build the most efficient, sturdy bridges possible. In the world of digital communication, these "bridges" are called codes. They are mathematical structures designed to send messages across noisy channels (like a crowded radio or a shaky internet connection) without losing information.

This paper is about organizing and simplifying the search for the best possible bridges. The authors focus on a specific, complex type of bridge called Polycyclic Codes.

Here is the breakdown of their work using simple analogies:

1. The Problem: Too Many Blueprints

Imagine you have a massive library of blueprints for building bridges. Some blueprints look slightly different on paper, but if you build them, they turn out to be identical in strength and shape. They are just "renamed" or "rearranged" versions of the same structure.

In coding theory, these are called equivalent codes. If you find a great code, you don't want to waste time searching for its "twins" or "cousins" that are mathematically the same. You want to find the unique "master blueprint" for each family of bridges.

2. The New Tool: The "n-Equivalence" Filter

The authors introduce a special filter called n-equivalence. Think of this as a magical sorting machine.

How it works: It takes two different-looking blueprints (polynomials) and checks if you can transform one into the other just by stretching or shrinking the coordinates (multiplying by a number).
The Result: If the machine says "Yes," they are the same family. If "No," they are truly different.

This helps researchers stop looking at the whole messy library and instead focus only on the representative of each family. It's like saying, "We don't need to study every shade of red; let's just study the one 'pure red' that represents all the others."

3. The Specific Focus: Trinomial Codes

The paper zooms in on a specific, simpler type of blueprint called Trinomial Codes.

The Metaphor: Imagine a standard blueprint has hundreds of terms (like $x^{100} + x^{99} + ... + 1$ ). A Trinomial is a simplified version with only three terms (like $x^n - x^\ell - 1$ ).
Why it matters: These are the "skeletons" of the more complex codes. By understanding how to sort these simple three-term skeletons, the authors can figure out how to sort the complex, multi-term versions later.

4. The Big Discovery: The "Magic Number" Rule

The authors found a specific rule (a mathematical condition involving the greatest common divisor) that tells you when a complex trinomial code is actually just a "renamed" version of a very simple, standard code.

The Analogy: Imagine you have a locked box with a complex combination. The authors found that if the numbers in the combination meet a certain condition (specifically, if two numbers share no common factors other than 1), the box is actually just a standard, simple box in disguise.
The Payoff: This means that under certain conditions, any complicated trinomial code family is equivalent to a single, famous "canonical" code ( $x^n - x^\ell - 1$ ). This drastically reduces the number of blueprints researchers need to test.

5. The "Special Case" (pℓ-trinomials)

The paper also looks at codes where the length of the bridge is related to the "characteristic" of the field (a property of the mathematical universe they are working in, like how a clock works differently in base 12 vs. base 10).

They discovered that for these specific lengths, the complex blueprints break down into very predictable, repeating patterns (like a fractal or a tiled floor). This allows them to list exactly what the "best" versions of these codes look like.

6. The Real-World Application (in the paper's context)

The authors didn't just do the math; they used their new sorting machine to find optimal codes.

What they did: They applied their rules to specific examples (like codes of length 27 or 12).
The Result: They found specific blueprints that are "optimal." In bridge-building terms, this means they found codes that are as strong as mathematically possible for their size. They provided a list of these "champion" codes, which other engineers can now use immediately.

Summary

Think of this paper as a cataloging system for a massive warehouse of digital bridges.

The Problem: The warehouse is full of bridges that look different but are actually the same.
The Solution: The authors built a "n-equivalence" scanner that identifies the duplicates.
The Breakthrough: They proved that for a huge class of these bridges (trinomial codes), almost all of them are just variations of one single, simple design.
The Outcome: They used this knowledge to find the absolute strongest bridges (codes) for specific sizes, saving future engineers from having to reinvent the wheel.

The paper does not discuss medical uses or future AI applications; it is strictly about organizing mathematical structures to find the most efficient ways to send data.

Technical Summary: Equivalence of Families of Polycyclic Codes over Finite Fields

Problem Statement
The paper addresses the classification and equivalence of families of polycyclic codes over finite fields $\mathbb{F}_q$ . While cyclic and constacyclic codes are well-understood as ideals in quotient rings $\mathbb{F}_q[x]/\langle x^n - \lambda \rangle$ , polycyclic codes generalize these structures to ideals in $\mathbb{F}_q[x]/\langle f(x) \rangle$ , where $f(x)$ is an arbitrary nonzero polynomial. A central challenge in coding theory is the construction of codes with optimal parameters (length $n$ , dimension $k$ , and minimum distance $d$ ). Since exhaustive computer searches for optimal codes are computationally intractable for large parameters, the authors aim to reduce the search space by systematically organizing polycyclic codes into equivalence classes. Specifically, the paper investigates when two ambient spaces (the quotient rings defining the codes) are isometric, thereby allowing researchers to restrict their search to representatives of these equivalence classes.

Methodology
The authors extend the concept of $n$ -equivalence, previously defined for constacyclic codes, to the broader context of polycyclic codes. The methodology proceeds in three stages:

Definition of Equivalence: Two families of polycyclic codes associated with polynomials $f(x) = x^n - \vec{a}(x)$ and $g(x) = x^n - \vec{b}(x)$ are defined as $n$ -equivalent if there exists an $\alpha \in \mathbb{F}_q^*$ such that the map $\Phi_\alpha: f(x) \mapsto f(\alpha x)$ induces an $\mathbb{F}_q$ -algebra isomorphism between the quotient rings $\mathbb{F}_q[x]/\langle g(x) \rangle$ and $\mathbb{F}_q[x]/\langle f(x) \rangle$ that preserves the Hamming distance.
Algebraic Characterization: The paper derives necessary and sufficient conditions for this equivalence. This involves analyzing the roots of binomial and trinomial equations in $\mathbb{F}_q$ $F_{q}$ . Key tools include:
- Lemmas regarding the degree of $\gcd(x^n - a, x^m - b)$ and the number of solutions to $x^n - a = 0$ .
- The use of Schur products (component-wise multiplication) to relate the coefficient vectors of equivalent polynomials.
- Group-theoretic analysis of cyclic subgroups generated by powers of primitive elements in $\mathbb{F}_q^* \times \mathbb{F}_q^*$ .
Specialization and Generalization:
- $\ell$ -Trinomial Codes: The study focuses first on codes associated with trinomials of the form $x^n - a_\ell x^\ell - a_0$ (where $0 < \ell < n$ ). The authors derive explicit formulas for the number of equivalence classes and conditions under which a trinomial code is equivalent to a canonical form $x^n - x^\ell - 1$ .
- $p\ell$ -Trinomial Codes: A specific analysis is conducted for codes of length $n = p^\ell + r$ (where $p$ is the characteristic of $\mathbb{F}_q$ ), utilizing properties of Artin-Schreier polynomials and trace functions to determine irreducible factorizations.
- General Polycyclic Codes: The results are generalized to arbitrary polycyclic codes defined by vectors $\vec{a}$ with $m$ non-zero entries.

Key Contributions and Results

Definition of $(n, \ell)$ -Equivalence: The paper formalizes an equivalence relation for $\ell$ -trinomial codes. It proves that two pairs of coefficients $(a_0, a_\ell)$ and $(b_0, b_\ell)$ are equivalent if and only if the polynomials $a_0 x^n - b_0$ and $a_\ell x^{n-\ell} - b_\ell$ share a common root in $\mathbb{F}_q^*$ .
Counting Equivalence Classes: The authors provide an explicit formula for the number of $n$ -equivalence classes for $\ell$ -trinomial codes:
$N_{(n,\ell)} = (q-1) \gcd(q-1, n, \ell)$
For general polycyclic codes with $m$ non-zero coefficients, the number of classes is given by:
$N = \frac{(q-1)^m}{\text{lcm}_{0 \le j \le m-1} \left( \frac{q-1}{\gcd(n-i_j, q-1)} \right)}$
Reduction to Canonical Forms:
- It is proven that if $\gcd(n, n-\ell) = 1$ , any $\ell$ -trinomial code family is equivalent to a family associated with the polynomial $x^n - x^\ell - 1$ .
- For the case where $\gcd(n, n-\ell) \neq 1$ , the paper identifies specific representatives (involving powers of a primitive element $\xi$ ) that cover all equivalence classes.
Factorization of Specific Trinomials: The paper analyzes the factorization of trinomials of the form $x^{p^\ell+1} - x^{p^\ell} - 1$ and $x^{p^\ell+s} - x^{p^\ell} - 1$ . It establishes conditions under which these polynomials split into irreducible factors (often involving Artin-Schreier polynomials $x^p - x - \beta$ ), which directly informs the possible generator polynomials for codes in these families.
Construction of Optimal Codes: By restricting the search to representatives of the identified equivalence classes, the authors construct several examples of optimal polycyclic codes. Notable examples include:
- A $[27, 3, 18]_3$ code and a $[81, 3, 55]_3$ code derived from $3\ell$ -trinomial codes over $\mathbb{F}_3$ .
- Optimal codes over $\mathbb{F}_9$ and $\mathbb{F}_4$ with parameters matching or attaining known bounds (e.g., Griesmer bound).

Significance and Claims
The paper claims that its primary significance lies in providing a systematic framework for organizing and classifying polycyclic codes. By establishing explicit criteria for equivalence, the work reduces the complexity of searching for codes with optimal parameters. Instead of investigating all possible coefficient combinations, researchers can focus on a finite set of representatives for each equivalence class.

Furthermore, the authors note that understanding the equivalence of ambient spaces is crucial for the study of duality in polycyclic codes. Since the dual of a polycyclic code is not necessarily polycyclic with respect to the same ambient space but is often equivalent to one arising from a different space, this classification provides a foundation for future theoretical results on duality. The paper concludes that these results extend the successful application of equivalence relations from constacyclic codes to the more general and complex domain of polycyclic codes, offering a pathway to constructing infinite families of codes with near-optimal parameters.