# P2618: An extremal Type II binary code of length 72

- ID: `P2618`
- Reference: `extremal-type-ii-code-72`
- Page: https://theoremdb.org/statements/P2618
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist a binary self-dual doubly-even code with parameters \([72,36,16]\)?

### Definitions

- **Definition.** Self-dual means C equals its orthogonal complement under the standard binary inner product.
- **Definition.** Doubly even means every codeword has weight divisible by four. Minimum distance 16 makes a length-72 Type II code extremal.

### What counts as a solution

- Provide a 36 by 72 generator matrix, verify self-duality and doubly-even weights, and certify minimum distance 16; or prove that no such code exists.

## Status

The audit found no accepted construction or nonexistence proof for an extremal Type II code of length 72. [4](#reference-4) [5](#reference-5) [6](#reference-6)

## Work

### Evidence for the current status

**Proposition 1 (Existence remains open).** The audit found no accepted construction or nonexistence proof for an extremal Type II code of length 72.

Sloane posed this exact question in 1973. A Type II code of length 72 has dimension 36, and the Mallows-Sloane bound gives minimum distance at most 16. The requested code would attain that bound.

A 2025 primary preprint on small automorphism groups says that the best known minimum distance for binary self-dual length-72 codes is 12 and that the minimum-distance-16 question remains unsolved. Recent construction work continues to produce Type I and Type II \([72,36,12]\) codes. The primary-source audit through 2026-07-25 found no refereed construction of a \([72,36,16]\) code and no unrestricted exclusion.

The answer is recorded as open. The forced enumerator and automorphism results below are necessary conditions on a witness. Neither condition resolves the unrestricted case.

### Background and intake notes

The forced weight enumerator gives strong consistency checks for any construction and useful pruning counts for search certificates.

- Original intake status: A 2012 primary paper describes existence as a long-standing open problem and excludes automorphisms of order 6. Current status remains unverified.
- Search systematic generator matrices [I|A] with AA^T=I, quotienting coordinate permutations before minimum-distance testing. Preserve the exact automorphism restriction for every negative run.
- Trap: exclusions for automorphisms of a given order do not combine into unrestricted nonexistence unless all possible automorphism groups, including the trivial group, are covered.

- Recorded example: Any witness must contain exactly 249849 codewords of weight 16.

### Other known results

- **Computation 1** (reproduced): Gleason's theorem forces 249,849 words of weight 16 and fixes every other weight count. [5](#reference-5)
- **Proposition 2** (supported): A witness has automorphism group C1, C2, C3, C2 by C2, or C5. [2](#reference-2) [7](#reference-7) [8](#reference-8) [9](#reference-9) [10](#reference-10) [11](#reference-11) [3](#reference-3)

### Prior approaches

- **Route 1** (supported): Structured searches reach distance 12 or exclude selected symmetry classes; the unrestricted distance-16 question survives. [3](#reference-3) [6](#reference-6) [10](#reference-10) [9](#reference-9)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python expands the forced polynomial and checks all coefficients, symmetry, size, distance, and design divisibilities. [5](#reference-5)

### Computational notes

- An independent Gleason-basis calculation obtained W=A^9-126A^6B+3015A^3B^2-4398B^3. Its nonzero coefficients begin A_16=249849, A_20=18106704, A_24=462962955 and are all nonnegative; the enumerator test therefore leaves existence open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `extremal-type-ii-code-72`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Martino Borello, “The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6”. DOI 10.1109/TIT.2012.2211095. arXiv:1203.3321 (2012). Generated after checking Borello, The Automorphism Group of an Extremal [72,36,16] Code Does Not Contain Elements of Order 6, 2012 https://arxiv.org/abs/1203.3321
   - preprint; primary source; arXiv:1203.3321v2; checked 2026-08-01
   - Source use: original_summary
   - For An extremal Type II binary code of length 72: CC0 candidate record based on a primary paper and an independent Gleason-polynomial calculation.
2. <a id="reference-2"></a>Martino Borello, The automorphism group of a self-dual [72,36,16] code is not an elementary abelian group of order 8, Finite Fields and Their Applications 25 (2014), 1-7. Martino Borello, Finite Fields and Their Applications 25 (2014), 1-7, Theorem 1.1 and final exclusion; combined with Vassil Yorgov and Daniel Yorgov, IEEE Transactions on Information Theory 60(6) (2014), 3302-3307, DOI 10.1109/TIT.2014.2313697 https://doi.org/10.1016/j.ffa.2013.07.007
   - Also cited at Martino Borello, Finite Fields and Their Applications 25 (2014), 1-7
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes an elementary abelian automorphism group of order eight.
3. <a id="reference-3"></a>Vassil Yorgov and Daniel Yorgov, The automorphism group of a self-dual [72,36,16] code does not contain Z4, IEEE Transactions on Information Theory 60(6) (2014), 3302-3307. Yorgov and Yorgov 2014, parameterization and exhaustive order-4 search; supplementary run materials at https://users.pfw.edu/yorgovd/code3/index.html; construction comparison from DOI 10.1007/s40314-024-03056-z https://doi.org/10.1109/TIT.2014.2313697
   - Also cited at Vassil Yorgov and Daniel Yorgov, IEEE Transactions on Information Theory 60(6) (2014), 3302-3307
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes cyclic automorphisms of order four from a length-72 extremal code.
4. <a id="reference-4"></a>Carolin Hannusch and Sándor Roland Major, On Small Automorphism Groups of Binary Self-Dual Codes, Preprints.org 202506.2100, version 1 (2025). Version 1, Section 6; the page labels the manuscript unreviewed https://www.preprints.org/manuscript/202506.2100
   - Also cited at On Small Automorphism Groups of Binary Self-Dual Codes, 2025, Section 6
   - preprint; primary source; version 1, posted 2025-06-25; checked 2026-07-25
   - Source use: citation_only
   - Reports, in an unreviewed manuscript, a reduction to five possible automorphism groups for any length-72 extremal code.
5. <a id="reference-5"></a>N. J. A. Sloane, Is there a (72,36) d=16 self-dual code?, IEEE Transactions on Information Theory 19(2) (1973), 251. N. J. A. Sloane, Is There a (72,36) d=16 Self-Dual Code?, IEEE Transactions on Information Theory 19(2) (1973), 251 https://doi.org/10.1109/TIT.1973.1054975
   - Also cited at Sloane 1973, weight-distribution table and 5-(72,16,78) consequence; independently expanded and checked in etc72-artifact-enumerator-replay
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Introduces the existence question for an extremal doubly even self-dual code of length 72.
6. <a id="reference-6"></a>Cong Yu, Shixin Zhu, and Tingting Wu, New binary [72,36,12] self-dual codes from group rings and skew group rings, Computational and Applied Mathematics 44(1) (2025), Article 1. Cong Yu, Shixin Zhu, and Tingting Wu, New binary [72,36,12] self-dual codes from group rings and skew group rings, Computational and Applied Mathematics 44 (2025), construction results https://doi.org/10.1007/s40314-024-03056-z
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Constructs new self-dual length-72 codes of minimum distance 12 and gives no distance-16 witness.
7. <a id="reference-7"></a>Stefka Bouyuklieva, On the automorphisms of order 2 with fixed points for the extremal self-dual codes of length 24m, Designs, Codes and Cryptography 25(1) (2002), 5-13. Stefka Bouyuklieva, Designs, Codes and Cryptography 25 (2002), 5-13, exclusion of involutions with fixed points https://doi.org/10.1023/A:1012598832377
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes fixed-point involutions from the automorphism group of a length-72 extremal code.
8. <a id="reference-8"></a>Stefka Bouyuklieva, On the automorphism group of a doubly-even (72,36,16) code, IEEE Transactions on Information Theory 50(3) (2004), 544-547. Stefka Bouyuklieva, IEEE Transactions on Information Theory 50(3) (2004), 544-547, exclusion of order-three automorphisms with fixed points https://doi.org/10.1109/TIT.2004.825252
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes order-three automorphisms with fixed points from a length-72 extremal code.
9. <a id="reference-9"></a>Thomas Feulner and Gabriele Nebe, The automorphism group of an extremal [72,36,16] code does not contain Z7, Z3 x Z3, or D10, IEEE Transactions on Information Theory 58(11) (2012), 6916-6924. Thomas Feulner and Gabriele Nebe, IEEE Transactions on Information Theory 58(11) (2012), 6916-6924, exclusions of C7, C3 by C3, and D10 https://doi.org/10.1109/TIT.2012.2208176
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes C7, C3 by C3, and D10 from the possible automorphism groups.
10. <a id="reference-10"></a>Martino Borello, The automorphism group of a self-dual [72,36,16] binary code does not contain elements of order 6, IEEE Transactions on Information Theory 58(12) (2012), 7240-7245. Martino Borello, IEEE Transactions on Information Theory 58(12) (2012), 7240-7245, exclusion of elements of order 6 https://doi.org/10.1109/TIT.2012.2211095
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes automorphisms of order six from a length-72 extremal code.
11. <a id="reference-11"></a>Martino Borello, Francesca Dalla Volta, and Gabriele Nebe, The automorphism group of a self-dual [72,36,16] code does not contain S3, A4, or D8, Advances in Mathematics of Communications 7(4) (2013), 503-510. Martino Borello, Francesca Dalla Volta, and Gabriele Nebe, Advances in Mathematics of Communications 7(4) (2013), 503-510 https://doi.org/10.3934/amc.2013.7.503
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Excludes S3, A4, and D8 from the possible automorphism groups.