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Classification of ternary maximal self-orthogonal codes of length 25

This paper presents a complete classification of ternary maximal self-orthogonal codes of length 25, extending previous results that covered lengths up to 24.

Original authors: Makoto Araya, Masaaki Harada

Published 2026-05-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Makoto Araya, Masaaki Harada

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 a master architect trying to build the perfect set of blueprints. In the world of mathematics, specifically in a field called coding theory, these "blueprints" are called codes. They are used to send messages (like text or images) across the universe without them getting garbled by static or errors.

This paper is about a specific type of blueprint called a Ternary Maximal Self-Orthogonal Code. That sounds like a mouthful, so let's break it down into a simple story about a very strict club.

The Setting: A Club with Strict Rules

Imagine a club where members are made of numbers. This club has three specific rules:

The Language: Everyone speaks a language with only three words: 0, 1, and 2. (This is the "Ternary" part).
The Length: Every member must have exactly 25 numbers in their ID card. (This is the "Length 25" part).
The "Self-Orthogonal" Rule: This is the club's most important rule. It's like a "mirror test." If you take any two members of the club and compare their ID cards, they must be perfectly "out of sync" in a mathematical way. If you mix their numbers together, the result must always be zero. In plain English: No two members can be too similar. They must be distinct enough that they cancel each other out.

The Goal: Finding the "Maximal" Club

The authors of this paper wanted to find every possible version of this club that is Maximal.

Maximal means the club is as full as it can possibly be. You cannot add one single new member to the club without breaking the "mirror test" rule. If you try to add a new person, they would inevitably be too similar to someone already inside.

So, the mission was: How many different ways can you build this perfect, full club of 25-number members, where everyone is unique and follows the strict rules?

The Challenge: A Needle in a Cosmic Haystack

The number of possible combinations is astronomically huge. It's like trying to find every possible arrangement of a deck of cards, but the deck has 25 cards, and each card can be one of three colors.

To solve this, the authors didn't just guess. They used a clever construction method:

The Lego Analogy: They started with smaller, known clubs (codes of length 24).
The Extension: They tried to "lengthen" these smaller clubs by adding one extra number to the end of every member's ID card.
The Filter: They checked every single new possibility to see if it still followed the rules. If it did, they kept it. If it broke the rules, they threw it away.

They also used a "Mass Formula," which is like a mathematical accounting trick. It's a way to check their work. Imagine you have a giant jar of marbles. You don't count them one by one; instead, you weigh the jar and know exactly how many marbles are inside based on the weight of a single marble. The authors used this formula to prove they hadn't missed any clubs and hadn't counted any twice.

The Big Discovery

After running these calculations on supercomputers, the authors found the answer:

There are exactly 139,613 different ways to build this perfect club of length 25.

They didn't just stop at the total number; they categorized them by how "strong" the club is. In coding theory, "strength" is measured by the minimum weight (how many non-zero numbers a member has).

The Strongest Clubs: 26 of these clubs are incredibly robust (minimum weight 9).
The Medium Clubs: 118,984 clubs are of medium strength (minimum weight 6).
The Weakest Clubs: 20,603 clubs are the least robust but still valid (minimum weight 3).

Why Does This Matter?

The paper concludes that this work completes a massive puzzle. Mathematicians have been classifying these clubs for lengths 3 up to 24 for years. This paper fills in the final piece for length 25.

Think of it like a museum. For years, curators had displayed every unique artifact from sizes 3 through 24. This paper adds the entire collection of size 25 to the exhibit, ensuring the museum is now complete for that specific range.

The authors also looked ahead, using their math to guess how many clubs might exist for lengths 26, 27, 28, 29, and 30. They found that the numbers get huge very quickly (billions and trillions), suggesting that classifying those future sizes will be an even bigger challenge.

In summary: This paper is a definitive catalog. It tells us exactly how many unique, rule-abiding, maximal groups of 25-number codes exist, completing a long-standing mathematical classification project.

Technical Summary: Classification of Ternary Maximal Self-Orthogonal Codes of Length 25

Problem Statement
The paper addresses the classification of ternary maximal self-orthogonal codes. A ternary code $C$ over the field $\mathbb{F}_3$ is self-orthogonal if $C \subseteq C^\perp$ . It is termed maximal if it is not properly contained in any other self-orthogonal code. While maximal self-orthogonal codes of lengths up to 24 had been previously classified, the case for length $n=25$ remained open. Specifically, the authors aim to classify all inequivalent ternary maximal self-orthogonal codes of length 25, which necessarily have dimension $k=12$ (since $n$ is odd, the dimension is $(n-1)/2$ ).

Methodology
The classification employs a constructive approach based on the operations of shortening and lengthening, combined with rigorous equivalence testing and mass formula verification.

Lengthening Construction: The authors utilize the inverse of the shortening operation. They construct ternary maximal self-orthogonal $[25, 12]$ codes by extending ternary self-orthogonal $[24, 11]$ codes. Given a generator matrix for a $[24, 11]$ code, a new column vector $b$ is appended to form a $[25, 12]$ code. The vector $b$ must satisfy specific weight and orthogonality conditions: $\text{wt}(b) \ge d-1$ , $\text{wt}(b) \equiv 2 \pmod 3$ , and $Ab^T = 0$ , where $A$ is the parity part of the generator matrix. To avoid redundancy, only vectors where the first non-zero component is 1 are considered.
Source Codes: The construction relies on two distinct sources:
- Dual Distance 1: Codes derived directly from known ternary self-dual $[24, 12]$ codes by appending a zero coordinate.
- Dual Distance > 1: Codes derived from the complete classification of ternary self-orthogonal $[24, 11]$ codes. These subcodes were enumerated by extracting them from all inequivalent ternary self-dual $[24, 12]$ codes.
Equivalence Testing: To determine if two constructed codes are equivalent, the authors employ a graph isomorphism approach. They construct vertex-colored digraphs $\Gamma_{W(C)}(C)$ based on the codewords of specific weights (typically weight 9, 12, or 24). Isomorphism testing and automorphism group computation are performed using the software nauty and Magma.
Mass Formula Verification: The completeness of the classification is verified using the mass formula for ternary self-orthogonal codes. The sum of $2^n n! / |\text{Aut}(C)|$ over all found inequivalent codes is compared against the theoretical total $T(n, k)$ .

Key Results
The paper establishes the following theorem:

Theorem 1: There are exactly 139,613 inequivalent ternary maximal self-orthogonal $[25, 12]$ $[25, 12]$ codes.
- Minimum Weight Distribution:
  - 26 codes have minimum weight $d=9$ .
  - 118,984 codes have minimum weight $d=6$ .
  - 20,603 codes have minimum weight $d=3$ .
Sub-classification:
- The authors provide a complete classification of ternary self-orthogonal $[24, 11]$ codes, finding 1,373 codes with $d=9$ , 2,745,294 with $d=6$ , and 339,492 with $d=3$ .
- The mass formula was successfully verified for both the $[24, 11]$ and $[25, 12]$ classifications, confirming that no codes were missed.
Automorphism Groups: Detailed tables are provided listing the orders of the automorphism groups for the resulting $[25, 12]$ codes, categorized by their minimum weights.

Significance and Claims
The primary significance of this work is the completion of the classification of ternary maximal self-orthogonal codes for all lengths $n$ up to 25. The authors note that this result is a direct consequence of the prior classification of ternary self-dual codes of length 24.

Furthermore, the paper derives non-tight lower bounds for the number of inequivalent ternary maximal self-orthogonal codes for lengths $n=26$ through $n=30$ using the mass formula. Specifically, it establishes that $N(26) \ge 757,009,213$ and $N(30) \ge 4.9 \times 10^{15}$ .

Finally, the authors identify a specific application of their classification: it serves as a foundation for a future project to complete the classification of weighing matrices of order 25 and weight 9, utilizing methods previously applied to length 24. The full list of codes is made available electronically via a university repository.

The Setting: A Club with Strict Rules

The Goal: Finding the "Maximal" Club

The Challenge: A Needle in a Cosmic Haystack

The Big Discovery

Why Does This Matter?

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