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Reversible double cyclic codes over a chain ring

This paper investigates the structural properties, duals, and minimal spanning sets of double cyclic codes over the chain ring $\mathbb{F}_q + u\mathbb{F}_q$ ( $u^2=0$ ), establishing conditions for reversibility and reversible-complementarity to construct DNA codes and optimal codes over $\mathbb{F}_4 + u\mathbb{F}_4$ .

Original authors: Mohd Anwar, Mohd Arif Raza, Mohd Rashid, Muzibur Rahman Mozumder

Published 2026-06-05

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CC BY 4.0

Original authors: Mohd Anwar, Mohd Arif Raza, Mohd Rashid, Muzibur Rahman Mozumder

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 trying to send a secret message across a noisy room. Sometimes, the message gets garbled or parts of it get lost. In the world of computers and data, we use "codes" to fix these errors. This paper is about designing a very specific, high-tech type of code called a Double Cyclic Code.

Here is a breakdown of what the authors did, using simple analogies.

1. The Setting: A Special Kind of Lockbox

Usually, mathematicians build codes using simple number systems (like just 0s and 1s). This paper uses a slightly more complex "lockbox" called a Chain Ring (specifically $F_q + uF_q$ ).

Think of this ring like a two-layered sandwich:

The bottom layer is a standard number system.
The top layer is a special "ghost" layer (represented by $u$ ) that interacts with the bottom layer but has a unique rule: if you multiply the ghost layer by itself ( $u^2$ ), it disappears (becomes zero).

This structure allows for more complex patterns than simple 0s and 1s, giving the code more "muscle" to fight errors.

2. The "Double" Dance: Cyclic Shifts

The core of this paper is about Double Cyclic Codes.

Imagine you have two separate lines of dancers holding hands:

Line A has $\gamma$ dancers.
Line B has $\delta$ dancers.

In a normal "cyclic" code, if everyone in Line A takes one step to the right, the person at the end wraps around to the front. In a Double Cyclic code, both lines do this dance at the same time.

Line A shifts right.
Line B shifts right.
The code is "valid" only if, after this double dance, the new formation still looks like a valid message.

The authors figured out exactly how to build these formations. They found that every valid code can be built from a few "master patterns" (called generator polynomials). It's like saying, "If you know these three specific dance moves, you can create every possible valid formation in the room."

3. The Mirror Test: Reversibility

The paper also looks at Reversible Codes.

Imagine you write a word on a piece of paper. If you flip the paper over and look at it in a mirror, does it still look like a valid word?

Non-reversible: "DOG" becomes "GOD" (which is a different word).
Reversible: "MADAM" becomes "MADAM" (it's the same).

In coding, this is crucial because sometimes the receiver gets the message "backwards." If the code is reversible, the computer doesn't have to panic; it knows that the backward version is still a valid message. The authors figured out the exact mathematical rules (involving "self-reciprocal" polynomials) that guarantee a code will pass this mirror test.

4. The DNA Connection: The Watson-Crick Rule

The most exciting part of the paper is applying this to DNA.

DNA is nature's hard drive. It uses four letters: A, T, C, and G.

The Rule: A always pairs with T, and G always pairs with C. This is the "Watson-Crick" rule.
The Problem: If you store data in DNA, you have to make sure that if the DNA strand flips over (reverses) and swaps partners (complements), it doesn't accidentally look like a different, valid message. That would cause a data crash.

The authors used their "Double Cyclic" math to build DNA Codes.

They mapped their mathematical "sandwich" numbers to DNA letters (A, T, C, G).
They ensured that if you take a DNA message, reverse it, and swap the letters (A $\leftrightarrow$ T, C $\leftrightarrow$ G), the result is still a valid code in their system.

5. The Results: Building Better DNA Storage

The paper doesn't just talk theory; they built actual examples.

They created specific "recipes" (generator sets) for these codes.
They showed that these recipes produce optimal codes, meaning they are very efficient at storing data and correcting errors.
They provided tables of actual DNA sequences (strings of A, T, C, G) that follow these rules. For example, they showed how to create a DNA code of length 16 or 24 that is robust against errors.

Summary

In plain English, this paper is a blueprint for building a super-stable, double-layered dance routine for data.

They defined the rules for how two lines of data can shift together without breaking.
They figured out how to make sure the routine looks the same even if you watch it in a mirror (reversibility).
They translated these rules into DNA language, creating a new way to store digital information in biological molecules that is less likely to get corrupted when the strands twist and turn.

The authors didn't claim this will cure diseases or build robots today; they simply proved that these specific mathematical structures exist, how to construct them, and that they work perfectly for the specific constraints of DNA storage.

Technical Summary: Reversible Double Cyclic Codes Over a Chain Ring

Problem Statement
This paper addresses the structural characterization of double cyclic codes over the finite chain ring $R = \mathbb{F}_q + u\mathbb{F}_q$ with $u^2 = 0$ , where $q$ is a prime power. While cyclic codes and their generalizations (such as double cyclic codes) over various rings have been studied extensively, the specific properties of reversible and reversible-complement double cyclic codes over this particular ring structure remain under-explored. The authors aim to determine the algebraic structure of these codes, identify their minimal spanning sets, analyze their duals, and establish necessary and sufficient conditions for reversibility. A primary motivation for this investigation is the application of these codes to the construction of DNA codes, which require specific constraints such as reversibility and the Watson-Crick complement rule to ensure biological stability and prevent non-specific hybridization.

Methodology
The authors employ algebraic coding theory over finite commutative rings. The methodology proceeds through the following steps:

Algebraic Structure: The paper defines double cyclic codes of length $(\gamma, \delta)$ as submodules of $R[x]/\langle x^\gamma - 1 \rangle \times R[x]/\langle x^\delta - 1 \rangle$ . Assuming $\gamma$ and $\delta$ are coprime to $q$ , the authors utilize the structure of cyclic codes over $R$ (derived from Theorem 2.1) to decompose double cyclic codes into generators involving polynomials $g_i(x), a_i(x) \in \mathbb{F}_q[x]$ and a linking polynomial $t(x)$ .
Generator Analysis: The structure is formalized as $C = \langle (g_1(x) + ua_1(x)|0), (t(x)|g_2(x) + ua_2(x)) \rangle$ . The authors analyze the conditions under which these codes are separable (i.e., $t(x) = 0$ ) and derive constraints on the degrees and divisibility of the generator polynomials.
Dual Codes: Using a bilinear map $\phi$ and reciprocal polynomials, the authors derive the generator polynomials for the dual code $C^\perp$ .
Reversibility Conditions: The concept of reversibility is defined via the reverse of a codeword. The authors utilize properties of reciprocal polynomials ( $f^*(x)$ ) to establish conditions under which the reverse of any codeword remains in the code.
DNA Code Construction: The study extends to the ring $R = \mathbb{F}_4 + u\mathbb{F}_4$ . By mapping ring elements to DNA base pairs (Adenine, Guanine, Thymine, Cytosine) via a specific correspondence $\tau$ , the authors construct DNA codes. They enforce the "reversible-complement" property, which ensures that if a codeword is in the set, its reverse-complement is also present, a critical requirement for DNA computing applications.

Key Contributions and Results

Structure and Spanning Sets: The paper provides a complete structural description of double cyclic codes over $R$ when lengths are coprime to $q$ . It identifies a minimal spanning set for these codes as an $R$ -module, consisting of four specific sets of vectors derived from the generator polynomials ( $N_1, N_2, N_3, N_4$ ). The cardinality of the code is explicitly calculated as $q^{2\gamma + 2\delta - r_1 - s_1 - r_2 - s_2}$ .
Duality: The generators for the dual code of a double cyclic code are determined in terms of the reciprocal polynomials of the original generators.
Reversibility Criteria:
- For separable codes, the code is reversible if and only if all generator polynomials ( $g_1, a_1, g_2, a_2$ ) are self-reciprocal.
- For non-separable codes (where $t(x) \neq 0$ ) and under specific length constraints ( $\delta = (kp+1)\gamma$ ), the authors derive a complex necessary and sufficient condition. This condition requires the generators to be self-reciprocal and imposes a divisibility constraint on the linking polynomial $t(x)$ involving its reciprocal $t^*(x)$ and specific ring elements.
Reversible-Complement Codes: The paper establishes that a double cyclic code is reversible-complement if and only if it is reversible and contains the vector $uI$ (where $I$ is the all-one vector).
DNA Code Construction: Using the ring $\mathbb{F}_4 + u\mathbb{F}_4$ , the authors construct specific DNA codes. They provide examples of double cyclic codes that satisfy the reversible-complement property and map them to DNA sequences. The paper lists constructed DNA codes of lengths 16 and 24, demonstrating minimum distances and codeword counts that align with optimal parameters found in existing databases (e.g., Grassl's database).

Significance and Claims
The authors claim that this work extends the existing literature on double cyclic codes from rings like $\mathbb{Z}_2$ , $\mathbb{Z}_4$ , and $\mathbb{F}_2 + u\mathbb{F}_2$ to the more general chain ring $\mathbb{F}_q + u\mathbb{F}_q$ . The significance lies in the rigorous algebraic characterization of reversibility and the reversible-complement property, which are essential for practical DNA computing applications.

The paper explicitly states that these codes are utilized to construct DNA codes that satisfy the Watson-Crick complement rule, Hamming distance constraints, and GC-content constraints. By providing explicit generator conditions and minimal spanning sets, the work facilitates the systematic generation of error-correcting codes suitable for DNA storage and computing. The authors conclude by noting that the construction of optimal codes (some marked with an asterisk in their tables) supports their theoretical results. They suggest that future work could explore these codes over different rings and lengths, further extending their application to DNA computing.