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Classification of LCD and self-dual codes over a finite non-unital local ring

This paper investigates LCD and self-dual codes over the noncommutative non-unital local ring $E_p$ by establishing conditions for MDS and AMDS properties and providing classifications of these codes for small lengths over $E_2$ and $E_3$ .

Original authors: Anup Kushwaha, Indibar Debnath, Om Prakash, Patrick Solé

Published 2026-06-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anup Kushwaha, Indibar Debnath, Om Prakash, Patrick Solé

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. To make sure the message arrives correctly, you add extra "guard" bits to your message. These guards help you spot if a letter got garbled (error detection) or even fix it if it's wrong (error correction). In the world of mathematics, these messages are called codes, and the rules they follow are like the grammar of a secret language.

For decades, mathematicians built these secret languages using a very specific, well-behaved type of number system called a field (think of it like a perfect, orderly grid of numbers). But recently, researchers started asking: "What if we use a messier, more chaotic number system?"

This paper explores exactly that. The authors are building secret codes using a strange, "messy" number system called $E_p$ .

The Setting: A Chaotic Number System

Think of the number system $E_p$ as a room with two special keys, $r$ and $s$ .

In a normal world, if you have a key, you can usually open any door. But in this room, there is no master key (no "multiplicative identity").
The rules are weird: If you turn key $r$ twice, it's the same as turning it once ( $r^2=r$ ). But if you turn $r$ then $s$ , you get $r$ ($rs=r$). If you turn $s$ then $r$ , you get $s$ ($sr=s$).
It's a non-commutative, non-unital ring. In plain English: The order in which you do things matters, and there's no "1" to multiply by.

The authors are trying to build the best possible secret codes using this chaotic room.

The Three Types of Codes They Studied

The paper focuses on three specific types of codes, each with a special superpower:

1. LCD Codes (The "Clean Break" Codes)

The Analogy: Imagine you have a team of spies (your code) and a team of counter-spies (the "dual" code). Usually, these two teams might overlap; a spy could be working for both sides.
The Goal: An LCD (Linear Complementary Dual) code is a team where the spies and counter-spies have zero overlap. They are completely separate.
Why it matters: In the real world (though the paper focuses on the math), having no overlap makes the code very secure against certain types of hacking attacks.
The Paper's Discovery:

The authors found a rule: To build a perfect LCD code in this chaotic room ( $E_p$ ), you just need to take a perfect LCD code from a normal, orderly room (a field $F_p$ ) and "translate" it using a specific key ( $r$ ).
They counted how many of these unique codes exist for short message lengths (up to 13 for the binary version, 10 for the ternary version).
They also found the "best" versions of these codes (called MDS and AMDS), which are the most efficient at fixing errors.

2. Left Self-Dual Codes (The "Mirror" Codes)

The Analogy: Imagine a code that is its own reflection. If you look at the code in a mirror, you see the exact same code.
The Goal: A Self-Dual code is one where the team of spies is identical to the team of counter-spies.
The Twist: Because our number system is messy (order matters), we have to be careful. Is it a "Left Mirror" or a "Right Mirror"?

Left Self-Dual: The code looks the same when mirrored from the left.
Right Self-Dual: The code looks the same when mirrored from the right.
The Paper's Discovery:
Left Side: They successfully built and classified the "best" (MDS/AMDS) Left Self-Dual codes for lengths up to 12. They found that these codes are just "translations" of perfect mirror codes from the orderly world.
Right Side: They hit a wall. They proved that perfect (MDS) Right Self-Dual codes cannot exist in this chaotic room. Furthermore, the "almost perfect" (AMDS) Right Self-Dual codes can only exist if the message is exactly 2 letters long. Anything longer breaks the rules.

3. Two-Sided Self-Dual Codes (The "Perfect Symmetry" Codes)

The Analogy: This is the ultimate code. It is its own reflection from both the left and the right. It is perfectly symmetrical.
The Paper's Discovery:

They proved that these codes can only exist if the message length is an even number (like 2, 4, 6). You can't have a perfectly symmetrical code with an odd number of letters in this system.
They also proved you can't have a "perfect" (MDS) code with a minimum distance of 1 (which would mean the code is very weak).
They classified the best of these codes for very short lengths (up to 6 for the binary version, 4 for the ternary version).

The Big Picture: What Did They Actually Do?

The authors didn't invent a new phone or a new encryption app. Instead, they did a massive inventory check of mathematical possibilities.

They mapped the territory: They created a catalog (tables in the paper) listing every unique, best-in-class code they could find for short message lengths in this specific chaotic number system.
They found the shortcuts: They proved that you don't need to reinvent the wheel. If you have a good code from a normal number system, you can easily turn it into a good code for this chaotic system.
They found the dead ends: They proved that certain types of codes (like perfect Right Self-Dual codes) are impossible in this system, saving other mathematicians from wasting time looking for them.

Summary

Think of this paper as a construction guide for a very specific, weird type of Lego set.

The bricks are strange and don't fit together the usual way.
The authors figured out how to build the strongest, most symmetrical towers (codes) possible with these bricks.
They listed exactly how many unique towers they could build for small sizes.
They also proved that some specific tower designs are impossible to build with these bricks at all.

The result is a foundational map for anyone who wants to build error-correcting codes using this particular strange number system.

Technical Summary: Classification of LCD and Self-Dual Codes over a Finite Non-Unital Local Ring

Problem Statement
Traditional coding theory has largely focused on codes as vector spaces over finite fields or modules over unital rings (primarily commutative). Recent research has shifted toward non-unital rings, yet literature remains sparse regarding codes over noncommutative, non-unital rings. This paper addresses the classification and enumeration of Linear Complementary Dual (LCD) and self-dual codes over a specific finite non-unital local ring, denoted as $E_p$ . The ring is defined by two generators $r$ and $s$ with relations $pr = ps = 0$, $r^2 = r$ , $s^2 = s$ , $rs = r$, and $sr = s$, where $p$ is a prime. The authors specifically investigate Maximum Distance Separable (MDS) and Almost MDS (AMDS) codes within this algebraic structure, a domain previously unexplored for non-unital rings.

Methodology
The study employs algebraic structural analysis and computational enumeration. The methodology proceeds through the following steps:

Algebraic Foundations: The authors define $E_p$ as a local ring with a unique maximal ideal $I = \{mt \mid 0 \le m < p\}$ , where $t = r + (p-1)s$ . They establish a reduction map $\alpha: E_p \to E_p/I \cong \mathbb{F}_p$ .
Code Decomposition: For any $E_p$ -linear code $C$ , the authors utilize the residue code $Res(C)$ and the torsion code $Tor(C)$. A key structural result established is that a code $C$ is free if and only if $Res(C) = Tor(C)$.
Duality and LCD Construction: The paper defines left and right duals ( $C^{\perp L}$ and $C^{\perp R}$ ) based on the Euclidean inner product. It is proven that no non-zero right LCD codes exist over $E_p$ , restricting the study to left LCD codes. The authors demonstrate that an $E_p$ -LCD code is always free and is generated by a matrix of the form $rG$, where $G$ generates an LCD code over the base field $\mathbb{F}_p$ .
Monomial Equivalence: The study establishes that two free $E_p$ -linear codes are monomial equivalent if and only if their residue codes are monomial equivalent over $\mathbb{F}_p$ . This allows the enumeration of $E_p$ codes to be reduced to the enumeration of $\mathbb{F}_p$ codes.
MDS/AMDS Characterization: Necessary and sufficient conditions are derived showing that a free $E_p$ -linear code is MDS (or AMDS) if and only if its residue code is MDS (or AMDS) over $\mathbb{F}_p$ .
Computational Enumeration: Using the MAGMA computer algebra system and existing classifications of binary and ternary LCD/self-dual codes (citing works by Araya, Harada, and others), the authors enumerate and classify specific codes for lengths up to 13 (for LCD) and 12 (for left self-dual) over $E_2$ and $E_3$ .

Key Contributions and Results

LCD Codes:
- Proved that all LCD codes over $E_p$ are free and that the number of monomial inequivalent LCD codes over $E_p$ equals the number over $\mathbb{F}_p$ for the same length.
- Provided exact counts of monomial inequivalent LCD codes over $E_2$ (lengths up to 13) and $E_3$ (lengths up to 10).
- Classified all MDS and AMDS LCD codes over $E_2$ and $E_3$ for lengths up to 6, providing explicit generator matrices.
Left Self-Dual Codes:
- Established that a left self-dual code over $E_p$ is free and its residue code is a self-dual code over $\mathbb{F}_p$ .
- Classified MDS and AMDS left self-dual codes over $E_2$ and $E_3$ for lengths up to 12.
- Demonstrated that MDS and AMDS properties for left self-dual codes over $E_p$ are directly inherited from their residue codes over $\mathbb{F}_p$ .
Right Self-Dual Codes:
- Proved the non-existence of MDS right self-dual codes over $E_p$ .
- Showed that AMDS right self-dual codes exist if and only if the code length is 2.
Two-Sided Self-Dual Codes:
- Proved that any MDS or AMDS self-dual code over $E_p$ must have an even length.
- Showed that no MDS self-dual code exists with minimum distance $d=1$ .
- Classified MDS and AMDS self-dual codes over $E_2$ (lengths up to 6) and $E_3$ (lengths up to 4).

Significance
The paper claims to be the first attempt to study MDS and AMDS LCD and self-dual codes over a non-unital ring. By extending the theory from unital and commutative rings to the non-unital, noncommutative ring $E_p$ , the work fills a gap in the literature regarding coding over non-unital structures. The results provide a complete classification for short lengths, offering a foundational framework for future investigations into error-correcting codes over similar algebraic structures. The work relies on the correspondence between codes over $E_p$ and codes over $\mathbb{F}_p$ , effectively leveraging existing field-based classifications to solve the ring-based problem.