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Self-dual double cyclic codes over $\mathbb{F}_q$

This paper investigates self-dual double cyclic codes over finite fields by establishing necessary and sufficient conditions for their generation, analyzing their existence across specific length configurations, and providing explicit construction methods alongside connections to other self-dual code classes.

Original authors: Ricky Aditya, Aleams Barra, Djoko Suprijanto

Published 2026-07-30

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Ricky Aditya, Aleams Barra, Djoko Suprijanto

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 sending a secret message across a noisy radio channel. To make sure the message arrives without getting scrambled by static, you add extra "check" bits, creating a safety net. In the world of mathematics, these safety nets are called codes. Some codes are special because they are cyclic, meaning if you take the message and shift every letter one spot to the left (wrapping the last letter around to the front), it still looks like a valid, safe message. This property makes them incredibly efficient for computers to process.

Now, imagine you have a message that is actually two different stories glued together side-by-side. A double cyclic code is like a safety net for these two-part messages. It has a superpower: if you shift the first part of the story and the second part of the story independently, the whole thing remains a valid code. But the real magic happens when a code is self-dual. Think of a self-dual code as a perfect mirror image of itself. If you take the code and look at its "shadow" (its dual), the shadow is identical to the original. These codes are prized by mathematicians because their perfect symmetry often leads to the strongest, most efficient ways to protect data, and they connect to deep patterns in geometry and design.

The paper you are about to read dives into the world of these "two-part, mirror-image" codes. The authors, Ricky Aditya, Aleams Barra, and Djoko Suprijanto, are like master architects who have figured out the exact blueprints needed to build these codes. They didn't just guess; they proved exactly when these structures can exist and how to build them. They discovered that you can't just build them anywhere; the "ground" you build on (the specific mathematical field you use) and the "size" of your message (the length of the code) have to follow strict rules. For instance, if you want to build a code where both parts are the same length, you need a special kind of number system where -1 is a perfect square. If the parts are different lengths, the rules change again. The authors provide a recipe book: if you follow their steps and check their conditions, you can construct these perfect codes, and in many cases, they turn out to be the very best possible codes for their size.

The Blueprint for Perfect Mirror Codes

In the world of error-correcting codes, the goal is always to pack as much information as possible while keeping the safety net tight enough to catch errors. The authors of this paper focus on a specific, tricky type of code: the self-dual double cyclic code. To understand what they found, let's break down the ingredients.

First, a double cyclic code is like a two-lane highway. A message travels down the road, split into a left lane (length $r$ ) and a right lane (length $s$ ). The rule is that if you shift the cars in the left lane, the whole traffic pattern stays valid. If you shift the cars in the right lane, it also stays valid. Usually, these two lanes are independent, but in a double cyclic code, they are linked together in a specific algebraic way.

Second, self-dual means the code is its own twin. In math terms, if you take every possible message in the code and check it against every other message to see if they are "orthogonal" (a fancy way of saying they don't interfere with each other), the set of messages that pass this test is exactly the same set you started with. It's like a lock that is identical to its own key.

The paper's main job is to answer two big questions: When can we build these codes? and How do we build them?

The Rules of the Game

The authors start by establishing the rules for the "generating elements" of these codes. Think of these generators as the master keys that create the entire code. They proved that for a double cyclic code to be self-dual, these master keys must satisfy three very specific mathematical conditions involving polynomials (which are just algebraic expressions with variables like $x$ ).

One of the most exciting findings is that not every combination of numbers works. The paper explicitly rules out the idea that you can build these codes in just any number system.

The "Same Length" Case ( $r, r$ ): If you want to build a code where both lanes are the same length, you can only do it if your number system (the field $\mathbb{F}_q$ ) allows $-1$ to be a perfect square. For example, in the field of numbers modulo 5, $2^2 = 4$ , which is $-1$ (since $4+1=5$ ). So, you can build these codes there. But in the field modulo 3, no number squared gives $-1$ (which is 2). The paper proves you cannot build a self-dual double cyclic code of equal length in that system.
The "Double Length" Case ( $r, 2r$ or $2r, r$ ): If one lane is twice as long as the other, the rules get even stricter. The paper shows that for these to exist in odd-numbered systems, $-2$ must be a perfect square, and the shorter length $r$ must be divisible by 4. If these conditions aren't met, the code simply cannot exist.

The Construction Kit

Once the authors figured out the "when," they gave us the "how." They provided a step-by-step construction method.

Find the right polynomials: You need to find specific algebraic expressions that fit the conditions mentioned above.
Mix them together: Use these expressions to create two pairs of polynomials.
Generate the code: These pairs act as the seeds. When you grow them using the rules of the code, they produce the full self-dual double cyclic code.

The authors didn't just stop at theory; they built several examples to prove their recipe works.

Example 1: They built a code of length $(7, 7)$ over the field with 2 numbers (binary). They found that by choosing specific polynomials, they created a code that is not only self-dual but also optimal. This means it has the best possible ability to detect and correct errors for its size.
Example 2: They constructed codes over the field with 5 numbers, creating codes of length $(5, 5)$ . Again, these turned out to be optimal, meaning they are the strongest possible shields for data of that size.
Example 3: They even tackled the trickier case where the lengths are different, like $(8, 16)$ over the field with 3 numbers. By following their specific recipe involving polynomials that satisfy the $-2$ square rule, they successfully built these codes, which also achieved optimal error-correcting distances.

The Hidden Connections

One of the paper's most playful discoveries is how these codes are related to other types of codes.

The authors point out that a double cyclic code of length $(r, r)$ is mathematically equivalent to a 2-quasi-cyclic code of length $2r$ . It's like realizing that a two-lane highway is actually just a single, wider road with a different traffic pattern.
Similarly, codes of length $(r, 2r)$ are deeply connected to negacyclic codes. This connection is a powerful tool because it means that if you know how to build one type of code, you can use the authors' findings to build the other.

The Bottom Line

This paper is a definitive guide. The authors didn't just suggest that these codes might exist; they proved the exact conditions required for their existence. They showed that while these codes are powerful and often optimal, they are picky about where they can live. They require specific number systems and specific lengths.

For anyone trying to design the most efficient data protection systems, this paper is a treasure map. It tells you exactly which islands (number systems) to visit and what tools (polynomials) to bring. If you follow the map, you can build codes that are not only mathematically beautiful (being their own mirror images) but also practically perfect for keeping our digital world safe. The authors have turned a complex algebraic puzzle into a clear, actionable set of instructions, proving that even in the abstract world of finite fields, there are strict, beautiful rules that govern the art of protection.

Technical Summary: Self-Dual Double Cyclic Codes over $\mathbb{F}_q$

Problem Statement
This paper investigates the structural properties, existence conditions, and construction methods for self-dual double cyclic codes over a finite field $\mathbb{F}_q$ . While classical cyclic codes are well-understood as ideals in the ring $\mathbb{F}_q[x]/\langle x^n - 1 \rangle$ , double cyclic codes generalize this concept to codewords of length $n = r + s$ , partitioned into two segments of lengths $r$ and $s$ , where each segment is invariant under a cyclic shift. A double cyclic code is defined as self-dual if it is equal to its dual code ( $C = C^\perp$ ). The primary challenge addressed is characterizing the necessary and sufficient conditions for the existence of such codes and providing explicit construction methods for specific length configurations, particularly distinguishing between separable and non-separable cases.

Methodology
The authors utilize an algebraic approach grounded in module theory over the polynomial ring $\mathbb{F}_q[x]$ .

Structural Characterization: The paper establishes that a double cyclic code of length $(r, s)$ is an $\mathbb{F}_q[x]$ -submodule of $F_{q,r,s} := \mathbb{F}_q[x]/\langle x^r - 1 \rangle \times \mathbb{F}_q[x]/\langle x^s - 1 \rangle$ . It is shown that any such code is generated by two pairs of polynomials, $(b(x), 0)$ and $(l(x), a(x))$ , where $b(x) \mid (x^r - 1)$ and $a(x) \mid (x^s - 1)$ .
Dual Code Analysis: By defining a specific orthogonality operation $\circ$ involving reciprocal polynomials, the authors derive the generating polynomials for the dual code $C^\perp$ in terms of the generators of $C$ .
Self-Duality Conditions: The condition $C = C^\perp$ $C = C^{⊥}$ is translated into a system of algebraic equations involving the generators $a(x), b(x), l(x)$ $a (x), b (x), l (x)$ and their reciprocal polynomials. Specifically, if $d(x) = \gcd(b(x), l(x))$ $d (x) = g cd (b (x), l (x))$ , the self-duality requires:
- $d(x)b^*(x) = x^r - 1$
- $a(x)a^*(x)\beta(x) = x^s - 1$ (where $b(x) = d(x)\beta(x)$ )
- $\beta(x)$ divides $1 - \lambda(x)\lambda^*(x)$ (where $l(x) = d(x)\lambda(x)$ )
Case-Specific Analysis: The paper analyzes three specific length configurations:
- Length $(r, r)$ : Investigating non-separable cases where the code is not a direct product of two cyclic codes.
- Length $(r, 2r)$ and $(2r, r)$ : Exploring connections to negacyclic codes.
- Length $(r, s)$ with $\gcd(r, s) = 1$ : Analyzing the coprime case.

Key Contributions and Results

General Characterization (Theorem 3.4): The paper provides necessary and sufficient conditions for a double cyclic code generated by $(b(x), 0)$ and $(l(x), a(x))$ to be self-dual. These conditions relate the generators to the factorization of $x^r - 1$ and $x^s - 1$ and impose constraints on the polynomial $\lambda(x)$ .
Length $(r, r)$ :
- Existence: A non-separable self-dual double cyclic code of length $(r, r)$ exists over $\mathbb{F}_q$ if and only if $-1$ is a square in $\mathbb{F}_q$ and there exists a non-constant polynomial $f(x)$ such that $f(x)f^*(x)$ divides $x^r - 1$ .
- Implication: This implies such codes do not exist over $\mathbb{F}_{p^i}$ where $p \equiv 3 \pmod 4$ and $i$ is odd.
- Equivalence: The authors note that double cyclic codes of length $(r, r)$ are equivalent to 2-quasi-cyclic codes of length $2r$ .
- Examples: Explicit constructions are provided for lengths $(7,7)$ over $\mathbb{F}_2$ , $(3,3)$ over $\mathbb{F}_4$ , and $(5,5)$ over $\mathbb{F}_5$ , with some identified as optimal codes.
Length $(r, 2r)$ and $(2r, r)$ :
- Characteristic 2: For fields of characteristic 2, such codes exist if and only if $r$ is even.
- Odd Characteristic: For fields of odd characteristic, existence requires three conditions: $-2$ is a square in $\mathbb{F}_q$ , $r$ is divisible by 4, and a specific polynomial factorization condition for $x^r - 1$ holds.
- Connection: These codes are shown to be related to self-dual negacyclic codes.
- Examples: Constructions are provided for lengths $(8, 16)$ and $(16, 8)$ over $\mathbb{F}_3$ .
Length $(r, s)$ with $\gcd(r, s) = 1$ :
- Existence: For coprime $r$ and $s$ , non-separable self-dual codes exist only if both $r$ and $s$ are odd. The construction relies on finding polynomials $d(x)$ and $a(x)$ such that $-\alpha d(x)d^*(x) = 1 + x + \dots + x^{r-1}$ and $\alpha a(x)a^*(x) = 1 + x + \dots + x^{s-1}$ for some $\alpha \in \mathbb{F}_q$ .
- Examples: Explicit codes are constructed for lengths $(9, 11)$ over $\mathbb{F}_3$ , $(7, 9)$ over $\mathbb{F}_4$ , and $(7, 9)$ over $\mathbb{F}_7$ .

Significance and Claims
The paper claims to provide a comprehensive framework for understanding self-dual double cyclic codes. Its primary significance lies in:

Unifying Characterization: Offering a general set of necessary and sufficient conditions for self-duality that applies to any length $(r, s)$ and any finite field $\mathbb{F}_q$ .
Explicit Constructions: Moving beyond theoretical existence to provide concrete construction methods and explicit polynomial generators for various specific cases.
Optimality: Demonstrating that the constructed codes in several examples achieve optimal minimum distances for their respective parameters.
Theoretical Connections: Highlighting the relationships between self-dual double cyclic codes and other established classes, specifically self-dual 2-quasi-cyclic codes (for length $(r, r)$ ) and self-dual negacyclic codes (for lengths $(r, 2r)$ and $(2r, r)$ ). The authors suggest these connections could theoretically aid in constructing other classes of codes.

The authors maintain a modest scope, focusing on the algebraic characterization and construction of these codes without proposing new experimental applications or future implications beyond the theoretical connections noted.

The Blueprint for Perfect Mirror Codes

The Rules of the Game

The Construction Kit

The Hidden Connections

The Bottom Line

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