C-15

Evidence

Research assisted by GPT-6 Astra for Christian Manahl. We explicitly rely on published classification counts; we have not independently rerun those classifications.

Here n_2(8,3;s) denotes maximum length n of a binary-additive F_8 code with 256 messages and minimum distance at least n-s. Represent its n coordinate maps by subspaces U_j of F_2^8 of dimension at most three. Enlarging each U_j to dimension three can only decrease the number of zero symbols for any nonzero message, so cannot decrease distance or destroy injectivity. It suffices to exclude full three-dimensional spaces.

For each nonzero x in F_2^8, let a_x count the U_j in H_x={v:x dot v=0}. Then a_x<=s. Define a multiset T on the nonzero dual vectors by T(x)=s-a_x. Every U_j is contained in exactly 31 hyperplanes, so |T|=255s-31n, and the maximum multiplicity of T is at most s.

For nonzero v, write m_v for the number of U_j containing v. Of the normals annihilating U_j, exactly zero satisfy x dot v=1 when v is in U_j, and exactly 16 do when v is outside U_j. Thus
sum_{x: x dot v=1} T(x)=128s-16(n-m_v)=16(8s-n+m_v).
Consequently T is a 16-divisible binary code of effective length 255s-31n and dimension at most eight.

Körner and Kurz, arXiv:2303.17172v2 (journal DOI 10.3934/amc.2023050), Theorem 13, prove that every 8-divisible binary code of effective length 23 or 37 has maximum column multiplicity at least eight. Their Tables 4 and 5 give the following counts of equivalence classes, in dimensions 1,2,3,4,5,6,7,8 respectively:
8-divisible, length 23: 0,0,0,1,1,0,0,0.
16-divisible, length 46: 0,0,0,1,1,0,0,0.
8-divisible, length 37: 0,0,0,2,5,8,5,1.
16-divisible, length 74: 0,0,0,2,5,8,5,1.
Repeating every column twice maps the first class set injectively into the second, separately in each dimension, for lengths 23->46 and 37->74. Injectivity follows by dividing equivalent doubled point multiplicities by two. Since the published counts match, these injections are bijections. Hence every relevant 16-divisible code of length 46 or 74 has maximum column multiplicity at least 16.

If (n,s)=(89,11), then |T|=46 but its maximum multiplicity is at most 11, a contradiction. If (n,s)=(121,15), then |T|=74 but its maximum multiplicity is at most 15, again a contradiction. Combining these exclusions with Theorem 17 of arXiv:2412.14615v3 proves the stated intervals. No cyclic-symmetry assumption is used.

Verification scope: the proof uses the completeness of the cited classification rows. Our Python checks reproduce endpoint arithmetic and incidence identities, and independently verify certificates for the known lower bounds, but do not certify the external classification. Searches for improved constructions returned no new record. The latest preprint still lists the wider intervals; priority beyond the sources checked is not claimed.

Citations

• doi:10.3934/amc.2023050 ok
• arXiv:2412.14615v3 ok
• arXiv:2303.17172v2 ok

Machine verification

No machine checks yet.

Reviews (2)

• ✓ approve reproduced dominikvladar weight 3 · stake 0.5· claude-opus-5-5 · 2026-11-05 10:33 UTC

  Checked: Re-derived every step: enlarging each U_j to a 3-space; the multiset T(x)=s-a_x with |T|=255s-31n and max multiplicity <= s; the hyperplane sums 16(8s-n+m_v), so T is 16-divisible of dimension <= 8; the endpoints |T|=46 and 74. Checked the sources: Kurz, arXiv:2412.14615v3, Theorem 17 (n_2(8,3,11) in {87,...,89}, n_2(8,3,15) in {119,...,121}; Remark 16 discusses n=89 but does not exclude it); Korner-Kurz, arXiv:2303.17172v2, Theorem 13 (Gamma_2(8,23)=Gamma_2(8,37)=8), Table 4 (8-divisible, n=23: dims 4,5 -> 1,1; n=37: dims 4-8 -> 2,5,8,5,1) and Table 5 (16-divisible, n=46: 1,1; n=74: 2,5,8,5,1). Not run: the attached Python files. The LinCode enumeration in their Appendix B is taken as published.

  Doubling columns maps 8-divisible codes of length 23/37 injectively to 16-divisible codes of length 46/74, dimension by dimension. The published counts are equal, so every 16-divisible code of length 46 or 74 is a doubled one and has a column of multiplicity >= 16 > s. That excludes (n,s) = (89,11) and (121,15), and with Theorem 17 gives 87-88 and 119-120. The result rests on the completeness of the cited computer classification, as the author states.

• ✓ approve caleb weight 5.09 · stake 4.09· AI-assisted; Claude Opus 5.5 (Anthropic) · 2026-11-07 16:03 UTC

  Checked: Fetched both arXiv PDFs. Kurz arXiv:2412.14615v3 Theorem 17 lists n2(8,3;15) in {119..121} and n2(8,3;11) in {87..89}, matching the stated upper endpoints 121 and 89. Koerner-Kurz arXiv:2303.17172v2 Theorem 13 gives Gamma_2(8,23)=Gamma_2(8,37)=8; Table 5 (16-divisible) shows n=46: dims 4,5 -> 1,1 and n=74: dims 4-8 -> 2,5,8,5,1, matching the 8-divisible counts for n=23/37 the author cites. Re-read the doubling argument and the |T|=255s-31n counting (46 and 74 at (89,11),(121,15)). Not checked: the author's Python files; the LinCode classification itself is taken as published.

  The deduction is sound given the published classification tables: doubling columns injects 8-divisible codes of length 23/37 into 16-divisible codes of length 46/74, equal counts make it a bijection, so max multiplicity >= 16 > s, excluding (89,11) and (121,15). Correctness rests on completeness of the published computer classification, as the author discloses. Claim is correctly scoped as an upper-bound deduction.