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Cyclic and Constacyclic Codes Over Z4+iZ4

This paper investigates cyclic and constacyclic codes over the finite chain ring $\mathbb{Z}_4+i\mathbb{Z}_4$ , proving their equivalence to cyclic codes, providing an algorithm for generating simple root constacyclic codes, and utilizing a Gray map to construct new best linear codes over $\mathbb{Z}_4$ .

Original authors: Miguel Martín (Faculty of Science,Engineering - Bernoulli Institute, University of Groningen), Ekin Özman (Faculty of Science,Engineering - Bernoulli Institute, University of Groningen)

Published 2026-07-22

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CC BY 4.0

Original authors: Miguel Martín (Faculty of Science,Engineering - Bernoulli Institute, University of Groningen), Ekin Özman (Faculty of Science,Engineering - Bernoulli Institute, University of Groningen)

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 room. In the world of digital communication, this "room" is the internet, and the "noise" is static that can scramble your words. To fix this, mathematicians create error-correcting codes. Think of these codes as a special way of packing your message into a suitcase. If the suitcase gets dropped and a few items get scrambled, the packing pattern is so clever that you can still figure out exactly what was inside. For a long time, scientists mostly used simple "on/off" switches (like zeros and ones) to build these suitcases. But in the 1990s, they discovered that using a slightly more complex set of four symbols (0, 1, 2, 3) could pack messages even tighter and protect them better. This is the world of quaternary codes.

Now, imagine you are a master packer who loves patterns. You notice that if you arrange your items in a circle and shift them one spot over, the pattern stays perfect. This is called a cyclic code. But what if you shift them and also twist them slightly? That's a constacyclic code. It's a more flexible, twisty version of the same idea. The big question for researchers has been: "Are these twisty codes actually just fancy versions of the simple circular ones, or are they totally different beasts?" And, more importantly, "Can we use these twisty patterns to build even better suitcases for our messages?" This is the puzzle a team of mathematicians set out to solve, exploring a strange, four-dimensional number system to see if they could unlock a new generation of super-secure digital messages.

In this paper, Miguel Martín and Ekin Özman dive into a specific, quirky number system called $R = \mathbb{Z}_4 + i\mathbb{Z}_4$ . If $\mathbb{Z}_4$ is a clock with only four hours (0, 1, 2, 3), then $R$ is like that clock but with a magical "imaginary" hand attached to it, creating a ring of 16 unique elements. The authors are hunting for constacyclic codes within this ring—patterns that stay intact even when you shift and twist the data.

The first major discovery they make is a bit like finding a secret shortcut in a maze. They prove that every single constacyclic code in this complex ring is actually equivalent to a standard cyclic code. In plain English, no matter how much you twist the pattern (using a "constacyclic shift"), you can always rearrange it to look like a simple, non-twisted circle. This is huge because it means researchers don't need to invent a whole new toolbox for these twisty codes; they can just use the tools they already have for simple cyclic codes. It turns a potentially messy problem into a clean, manageable one.

To find these codes, the authors had to break down complex mathematical "polynomials" (which are like algebraic recipes for building codes) into their simplest, indivisible parts. They developed a clever, step-by-step algorithm (a recipe for a computer) to do this. They started with a known factorization in a simpler world (a field with just two numbers) and used a technique called Hensel's Lift to "lift" those factors up into their complex ring. Think of it like taking a blueprint for a small house and using it to build a skyscraper, ensuring every floor fits perfectly. They wrote a computer program (using a tool called Magma) to run this algorithm for various lengths of codes, specifically looking at odd numbers up to 31.

Once they found these codes in the complex ring, they didn't stop there. They used a special "translation tool" called a Gray map to convert these codes into $\mathbb{Z}_4$ codes (the four-symbol codes mentioned earlier). This is the bridge that turns abstract math into practical data protection. By translating the codes, they could measure how well they would perform in the real world.

The result? They found new, better codes. Specifically, they discovered 15 new linear codes over $\mathbb{Z}_4$ that have better "Lee distance" (a measure of how well they can resist errors) than any previously known codes of the same size. For example, they found a code of length 30 that can handle more errors than any other known code of that length. They even found some codes that work for an infinite number of lengths, like a universal key that fits many different locks.

The authors are very sure of their findings because they didn't just guess; they proved the equivalence of the codes and computed the exact generators using rigorous mathematical algorithms. They explicitly ruled out the idea that these twisty codes are fundamentally different from cyclic ones in this specific ring, showing instead that they are just different faces of the same coin. While they found these codes through computer simulations and mathematical construction, the paper presents them as concrete, verified improvements to the database of known codes, ready to be used to make our digital communications more robust.

Technical Summary: Cyclic and Constacyclic Codes over $\mathbb{Z}_4 + i\mathbb{Z}_4$

Problem Statement
This paper investigates the structural properties of cyclic and constacyclic codes over the finite chain ring $R = \mathbb{Z}_4 + i\mathbb{Z}_4$ , where $i^2 = -1$ . While codes over the ring $\mathbb{Z}_4$ have been extensively studied due to their connections to non-linear binary codes and applications in bio-informatics and cryptography, this work focuses on the extension ring $R \cong \mathbb{Z}_4[u]/\langle u^2 + 1 \rangle$ . The primary objective is to characterize simple root constacyclic codes over $R$ (where the length $n$ is coprime to the characteristic of $R$ ) and to utilize these codes to generate new, high-performance linear codes over $\mathbb{Z}_4$ .

Methodology
The authors employ a combination of algebraic characterization, ring isomorphism, and computational algorithms:

Ring Structure Analysis: The paper establishes that $R$ is a finite commutative chain ring with a unique maximal ideal $\langle 1+i \rangle$ and residue field $k \cong \mathbb{F}_2$ . The units of $R$ are partitioned into two sets, $U_1$ and $U_2$ , based on the parity of their real components.
Equivalence Proof: A central theoretical result proves that every simple root $\lambda$ -constacyclic code over $R$ is equivalent to a cyclic code over $R$ . This is achieved by constructing a monomial transformation (a ring isomorphism) that maps the quotient ring $R[x]/\langle x^n - \lambda \rangle$ to $R[x]/\langle x^n - 1 \rangle$ . This transformation relies on the specific properties of units in $R$ and the parity of the code length $n$ .
Factorization via Hensel's Lift: To generate these codes, the authors develop an iterative algorithm to factor the polynomial $x^n - \lambda$ $x^{n} - λ$ in $R[x]$ $R [x]$ .
- First, $x^n - 1$ is factored over the intermediate ring $R/\langle 2 \rangle \cong \mathbb{F}_2 + i\mathbb{F}_2$ using the known factorization over $\mathbb{F}_2$ .
- An iterative lifting process (based on Hensel's Lemma) is then applied to lift these factors to $R[x]$ , ensuring the resulting polynomials are monic, pairwise coprime, and basic irreducible.
- For general units $\lambda$ , the factorization of $x^n - \lambda$ is derived from the factorization of $x^n - 1$ using a specific multiplicative map $T_\lambda$ .
Gray Map Construction: The study utilizes Gray maps to project codes from $R$ $R$ to $\mathbb{Z}_4$ $Z_{4}$ . Two maps are considered:
- $\phi: a+bi \mapsto (b, a+b)$ , which typically yields quasi-cyclic codes.
- $\phi': (x_1, \dots, x_n) \mapsto (b_1, a_1+b_1, \dots, b_n, a_n+b_n)$ , an alternative extension observed to sometimes yield cyclic $\mathbb{Z}_4$ codes.
- Lemma 5.2 establishes that if $C$ is a free $R$ -linear code of dimension $k$ , its image under these maps is a free $\mathbb{Z}_4$ -linear code of dimension $2k$ and length $2n$ .

Key Contributions and Results

Equivalence of Constacyclic and Cyclic Codes: The paper proves that for the ring $R$ , the study of constacyclic codes can be reduced to the study of cyclic codes, as they are monomially equivalent. This simplifies the search for codes with optimal parameters.
Algorithmic Generation: An algorithm is presented and implemented in Magma to compute the unique factorization of $x^n - \lambda$ for any odd $n$ and unit $\lambda$ . This allows for the systematic generation of all free simple root constacyclic codes over $R$ .
New Best Linear $\mathbb{Z}_4$ Codes: By applying the Gray maps to the constacyclic codes over $R$ $R$ , the authors obtained a set of linear $\mathbb{Z}_4$ $Z_{4}$ codes. These codes were compared against existing databases (e.g., [ALO23], [Ayd23]).
- Table 5.1 lists 15 new codes with parameters $[N, K, d_L]$ (length, dimension, minimum Lee distance) that improve upon previously known Lee distances for the same length and dimension.
- Table 5.2 identifies specific instances among these new codes that are themselves cyclic over $\mathbb{Z}_4$ under the map $\phi'$ , providing explicit generators for these cyclic codes.
Infinite Families: The paper identifies infinite families of cyclic $\mathbb{Z}_4$ codes generated by polynomials such as $x+3$ , $x^2+3$ , and $x^4+x^2+1$ , though it notes that those with Lee distance 2 are less significant for practical applications compared to the specific "best" codes found in Table 5.1.

Significance
The paper claims significance primarily in the expansion of the catalog of best-known linear codes over $\mathbb{Z}_4$ . By leveraging the algebraic structure of the extension ring $\mathbb{Z}_4 + i\mathbb{Z}_4$ , the authors demonstrate a method to discover codes with superior Lee distance parameters. The work reinforces the utility of finite chain rings as a source for constructing high-quality codes, extending the legacy of $\mathbb{Z}_4$ coding theory. The authors explicitly state that the new codes found have been added to the relevant databases, with the exception of a few cases where parameters tied with existing linear codes but offered a cyclic structure. The research is grounded in the established motivation that codes over extensions of $\mathbb{Z}_4$ are valuable both for their intrinsic properties and their ability to map to $\mathbb{Z}_4$ codes.

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