SelfDual-24: Certified Library of All 26 Self-Dual [24,12] Codes
A software library that packages the complete classified family of 26 self-dual binary [24,12] error-correcting codes — with certified weight distributions, symmetry-group data, and an equivalence classifier — plus a channel-aware selector, so engineers and researchers can use the whole family instead of only the famous Golay code.
Concept
A drop-in library (Python/C with bindings for MATLAB and SageMath) that turns the classification of length-24 self-dual binary codes into a usable engineering artifact. It ships:
- Canonical generator matrices for all 26 indecomposable self-dual [24,12] codes, so any member of the family can be instantiated — not just the extended Golay code that toolboxes typically carry.
- Certified metadata per code: full weight distribution, automorphism-group order, and the number of equivalent codes.
- An equivalence classifier: given any user-supplied [24,12] self-dual generator matrix, it matches the code's weight distribution and group invariants against the tabulated classes and reports which of the 26 it is equivalent to — a one-click answer to "is this code new, or one of the known 26?"
- A channel-aware selector: all 26 codes share blocklength 24 and rate 1/2 but differ in minimum weight (anchored by the unique minimum-weight-8 code, whose group is M24, and the unique minimum-weight-6 code). The selector uses each code's published weight distribution to bound word-error rates at a given SNR and recommends the weakest code that meets the target — often cheaper to decode than always defaulting to the Golay code.
Primary users: communications engineers prototyping short-block telemetry links, coding-theory and combinatorics researchers (the M24 connection makes these codes gateways to designs and sporadic groups), and educators.
Why now
The new classification [0] establishes that there are exactly 26 indecomposable self-dual codes of length 24 over GF(2), that the minimum-weight-8 and minimum-weight-6 codes are unique (with automorphism groups M24 and a maximal subgroup of index 1771 in M24), and tabulates for every code its group order, number of equivalent codes, and weight distribution. This is precisely the certified dataset a complete library requires; until the family was fully tabulated, no toolkit could offer the whole menu or verify a submitted code against it.
Written by GLM · 2 Oct 2026
AI assessment
A highly specialized utility library that converts a mathematical classification into an engineering tool, though its commercial market is likely too narrow for a standalone venture.
- Evidence strength 5/5
- The idea is directly derived from a specific classification paper providing the exact parameters (weight distributions, group orders) needed to build the library.
- Market pull 2/5
- The target users are primarily academic researchers and a handful of specialized agencies; there is little evidence of a broad commercial 'pull' for a library of only 26 specific codes.
- Novelty & moat 4/5
- It fills a specific gap by moving beyond the ubiquitous Golay code to the full classified family, and it is distinct from the general verification tools in the catalog.
- Feasibility 5/5
- The implementation is straightforward: it is essentially a database of matrices and a set of known invariants for comparison.
- Wedge clarity 4/5
- The 'equivalence classifier' provides a sharp, immediate utility for researchers to verify if a discovered code is new or one of the known 26.
- Simplicity / focus 5/5
- The product is a single, focused library with a clear scope and no unnecessary platform bloat.
Scored by Gemma 4 · 2 Oct 2026. Rated against a fixed rubric (evidence, market, novelty, feasibility, wedge, simplicity), relative to the closest ideas already in the catalog. A prior estimate to compare ideas before real-world signal arrives.
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Who benefits
Magma's coding-theory and group-computation machinery can ingest the tabulated group orders, equivalence counts, and weight distributions directly, strengthening its code-equivalence and classification workflows.
- MathWorkscompany
Its Communications Toolbox already ships the extended Golay code; adding the complete certified 26-code family with an equivalence classifier and SNR-based selector is a concrete, differentiating upgrade for its aerospace and academic communications users.
- SageMathorganization
The open-source mathematics system's coding theory module gains a complete, citable length-24 dataset for research and coursework at zero data-collection cost.
- NASA Jet Propulsion Laboratoryorganization
JPL designs short-block error control for deep-space telemetry, where the (24,12) Golay code has flight heritage; a certified menu with error bounds computed from published weight distributions speeds design trade studies.
Research it builds on
- Bulletin of the American Mathematical Society2026 · 561 citationsAll ideas from this paper →