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On the structure of constacyclic codes over finite chain rings

This paper presents an explicit construction for the minimum set of generators of arbitrary-length $\lambda$ -constacyclic codes over finite chain rings, derives their rank and minimal spanning sets, and establishes necessary and sufficient conditions for these codes to be Maximum Hamming Distance with respect to Rank (MHDR) or Maximum Distance Separable (MDS) based on their torsion codes over the residue field.

Original authors: Vaishali Singh, Sucheta Dutt, Ridhima Thakral

Published 2026-07-03

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CC BY 4.0

Original authors: Vaishali Singh, Sucheta Dutt, Ridhima Thakral

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 very noisy, chaotic room. To make sure the message arrives correctly, you wrap it in a special "error-correcting" package. In the world of mathematics, these packages are called codes.

This paper is like a master blueprint for building a specific, highly efficient type of these packages called constacyclic codes, but with a twist: instead of building them on simple, flat ground (like standard number systems), the authors are building them on a complex, multi-layered structure called a Finite Chain Ring (FCR).

Here is a simple breakdown of what the paper achieves, using everyday analogies:

1. The Building Blocks: The "Chain Ring"

Think of a Finite Chain Ring as a set of nested Russian dolls or a multi-story building.

The bottom floor is a simple field (like a basic set of numbers).
As you go up, each floor is built on the one below it, but with a special "glue" (represented by a symbol $\gamma$ ) that holds them together.
The paper deals with codes built on these multi-story structures, which allows for more complex and robust error correction than simple flat structures.

2. The "Constacyclic" Shift

Imagine your message is a necklace of beads.

A cyclic code is like a necklace where if you slide every bead one spot to the right, the last bead wraps around to the front, and it still looks like a valid necklace.
A constacyclic code is a slightly more flexible version. When you slide the beads, the last one doesn't just wrap around; it might get multiplied by a special "magic number" ( $\lambda$ ) before it snaps into place.
The paper focuses on finding the best way to construct these specific types of necklaces.

3. The Main Discovery: The "Minimal Toolkit"

The biggest problem the authors solved is: "What is the smallest, most efficient set of tools (generators) needed to build any of these codes?"

The Old Way: Sometimes, people tried to build these codes using a messy pile of tools, many of which were redundant (like using a hammer, a rock, and a heavy book to drive a nail).
The New Way: The authors created a step-by-step recipe to find the minimal set of generators.
- They look for the "shortest" polynomial (the simplest tool) first.
- Then they look for the next shortest one that adds something new.
- They keep going until they have the perfect, lean team of tools.
The Result: They proved that this specific team of tools is the smallest possible team needed to build the code. No extra tools are needed, and no tools are missing. They also calculated the exact "rank" (size) of the code based on this minimal team.

4. The "Torsion" Connection: Peeling the Onion

To understand if these complex codes are truly "perfect," the authors use a technique called looking at Torsion codes.

Imagine your complex code is a thick onion. The Torsion code is like peeling away the outer layers to look at the very core (the residue field).
The paper proves a powerful rule: If the core (the Torsion code) is a "perfect" code, then the whole onion (the complex code) is also a "perfect" code.
This allows mathematicians to check the quality of a complex, multi-layered code just by looking at its simple, flat core.

5. The "Perfect" Codes: MHDR and MDS

The paper defines two types of "perfect" performance for these codes:

MDS (Maximum Distance Separable): Think of this as the "Gold Standard." It means the code is as far apart from other possible messages as mathematically possible. It offers the maximum protection against errors.
MHDR (Maximum Hamming Distance with respect to Rank): This is a slightly different kind of "Gold Standard" that specifically accounts for the size of the toolkit (the rank) used to build it.

The Paper's Conclusion on Perfection:
The authors provide a clear checklist (necessary and sufficient conditions) to tell you exactly when a code will be MDS or MHDR.

For MHDR: You just need to check if the core (Torsion code) is perfect.
For MDS: It's stricter. The core must be perfect, AND the code must be built using a single, clean "principal" generator (like using one perfect master tool instead of a team of different tools).

Summary

In short, this paper is a construction manual for high-tech error-correcting codes.

It tells you exactly which tools you need to build them (no more, no less).
It tells you how to measure their size (rank).
It gives you a test to see if your code is the "best possible" (MDS or MHDR) by looking at its simple core.

The authors didn't just guess; they provided a mathematical proof that their method is the most efficient way to generate these codes and gave specific examples (like codes built on numbers modulo 125 or 343) to show how the recipe works in real life.

Technical Summary: On the Structure of Constacyclic Codes over Finite Chain Rings

Problem Statement
The paper addresses the structural characterization of $\lambda$ -constacyclic codes of arbitrary length $\ell$ over a finite chain ring (FCR) $R$ . While significant progress has been made in understanding cyclic and constacyclic codes over finite fields and specific lengths (such as prime powers or lengths coprime to the characteristic), a comprehensive explicit construction for generators of codes of arbitrary length over general FCRs, along with their minimal spanning sets and rank, remained an area requiring further development. Additionally, the paper seeks to establish necessary and sufficient conditions for these codes to achieve Maximum Hamming Distance with respect to Rank (MHDR) and Maximum Distance Separable (MDS) properties, extending previous results that were often limited to specific ring structures or code lengths.

Methodology
The authors employ algebraic techniques rooted in the theory of finite commutative chain rings and polynomial rings. The methodology proceeds through the following logical steps:

Algebraic Framework: The study utilizes the structure of an FCR $R$ , characterized by a unique maximal ideal $\langle \gamma \rangle$ with nilpotency index $\rho$ and a residue field $F_q$ . The ring $R[x]/\langle x^\ell - \lambda \rangle$ is identified as the ambient space where $\lambda$ -constacyclic codes correspond to ideals.
Generator Construction: The core method involves constructing a set of generators $\{f_0(x), f_1(x), \dots, f_s(x)\}$ for a code $C$ . This is achieved by iteratively selecting polynomials of minimum degree from the code (or the difference between the code and the ideal generated by previously selected polynomials). A crucial selection criterion is the power of $\gamma$ in the leading coefficient; specifically, the authors select polynomials such that the power of $\gamma$ in the leading coefficient strictly decreases ( $r_0 > r_1 > \dots > r_s$ ) while the degrees strictly increase ( $n_0 < n_1 < \dots < n_s$ ).
Torsion Codes: The analysis leverages the concept of torsion codes, $Tori(C)$, defined over the residue field $F_q$ . The paper establishes a direct relationship between the generators of the code over $R$ and the generator polynomials of these torsion codes.
Rank and Spanning Set Derivation: Using the constructed generators, the authors derive a minimal spanning set $S$ by taking shifts of the generators up to specific degrees determined by the differences in the degrees of consecutive generators.
MHDR and MDS Characterization: The conditions for a code to be MHDR or MDS are derived by relating the minimum distance and rank of the code over $R$ to the properties of its zero-th torsion code $Torr_0(C)$ over $F_q$ .

Key Contributions and Results

Explicit Generator Construction: The paper provides an explicit construction for a generating set of any $\lambda$ -constacyclic code of arbitrary length $\ell$ over an FCR. Theorem 3.1 establishes that the set $\{f_0(x), \dots, f_s(x)\}$ , constructed via the minimum degree and leading coefficient criteria, generates the code as an ideal.
Minimal Spanning Set and Rank: Theorem 3.4 defines a minimal spanning set $S$ for the code $C$ . A significant result is the determination of the rank of the code: $\text{Rank}(C) = \ell - n_0$ , where $n_0$ is the degree of the minimum degree polynomial $f_0(x)$ in the code. This formula provides a direct link between the algebraic structure of the generator and the dimension of the code.
MHDR Conditions: Theorem 3.9 proves that a $\lambda$ -constacyclic code $C$ over $R$ is MHDR if and only if its zero-th torsion code $Torr_0(C)$ is an MDS code over the residue field $F_q$ . This reduces the problem of checking the MHDR property over a complex ring to checking the MDS property over a finite field.
MDS Conditions: Theorem 3.10 provides necessary and sufficient conditions for a code to be MDS. It states that $C$ $C$ is MDS if and only if:
1. $Torr_0(C)$ is MDS over $F_q$ .
2. $C$ is principally generated by a monic polynomial (implying the code is generated by a single polynomial $f_0(x)$ with $s=0$ and $r_0=0$ ).
  This result clarifies that not all MHDR codes are MDS; the MDS property requires the code to be principally generated.
Specific Length Constraints: Corollary 3.13 extends known results for MDS codes over finite fields to codes over FCRs, specifying the allowable degrees $n_0$ for the generator polynomial based on the length $\ell = np^m$ .

Significance and Claims
The paper claims to extend the theoretical framework of coding theory over finite chain rings by generalizing results from specific lengths (such as $2^s$ or prime powers) to codes of arbitrary length. The authors highlight that their construction achieves the minimum possible number of generators, which is a refinement over previous studies that noted repeated-root codes are generally not principally generated.

The significance of the work lies in its ability to:

Provide a unified algebraic description of constacyclic codes over FCRs, regardless of the code length.
Offer a computable method to determine the rank and minimal spanning set of such codes.
Bridge the gap between codes over rings and codes over fields by establishing that the MHDR property is entirely dependent on the torsion code over the residue field.
Clarify the distinction between MHDR and MDS codes in this context, demonstrating that while MDS codes are a subset of MHDR codes, the converse is not true unless the code is principally generated.

The paper concludes by noting that these theoretical results are illustrated through concrete examples involving rings like $\mathbb{Z}_{125}$ , $\mathbb{Z}_{343}$ , and $\mathbb{Z}_{289}$ , validating the derived formulas for rank, spanning sets, and distance properties.