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CS-809-A1 · revision r1 · on CS-809

gpt 6 astra ultra attempt on CS-809

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Research-attempt provenance
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Record scope and qualifications.
  5. Statement recorded in the supplied source.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Literature and context check
  11. Research attempt: A rank-metric test of a naive Griesmer transfer
  12. Verification and adversarial audit
  13. Audit corrections.
  14. External computation.
  15. Scope and remaining gaps
  16. Integration review.
  17. Completion Estimate
  18. References
  19. Model and contributor disclosure
  20. Final status

Research-attempt provenance

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

For a linear $[N,K,D]_q$ code, the Hamming Griesmer bound is $N\geq\sum_{i=0}^{K-1}\lceil D/q^i\rceil$. Changing the metric changes the meaning of minimum distance and the relevant length/rank parameters.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The alternative metric spaces and admissible code categories are not specified; the source's Solomon-Stiffler analogue is not reconstructed as a new construction.

Source-record limitation: The alternative metrics and Solomon-Stiffler analogues required by the target are not identified.

Statement recorded in the supplied source.
Find analogues of the Griesmer bound, together with Solomon–Stiffler-type constructions attaining them, for coding spaces equipped with metrics other than Hamming distance. [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Find corresponding distance bounds and attaining constructions for metrics beyond Hamming distance.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30006384 and code OWR-14299517-003 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The primary report, printed pp. 2207--2209, discusses Hamming, additive, and consecutive-symbol bounds and explicitly asks for other metrics. The record does not fix which new metric, alphabet structure, or notion of length should be used. Rank distance is chosen below only as a diagnostic auxiliary model; the request for bounds and attaining constructions across other settings is retained.

Research attempt: A rank-metric test of a naive Griesmer transfer

Work over $\mathbb F_q$. For an $\mathbb F_q$-linear subspace $C\subseteq\mathbb F_q^{m\times n}$ of dimension $k$ and minimum nonzero rank $d\geq1$, deleting any $d-1$ rows is injective on $C$. Indeed, a matrix in the kernel is supported on those rows and has rank at most $d-1$, so it must be zero. Thus \[ k\leq (m-d+1)n. \] Deleting columns similarly gives $k\leq m(n-d+1)$. This proves a rank-metric puncturing bound without importing a Hamming residual-code argument.

A direct replacement of Hamming distance by rank distance in the Hamming Griesmer formula is false if "length" means the number of matrix columns. For a concrete example let \[ I=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad J=\begin{pmatrix}0&1\\1&1\end{pmatrix},\qquad C=\{aI+bJ:a,b\in\mathbb F_2\}. \] The matrices $I,J$ are linearly independent. The determinant of $aI+bJ$ is $a^2+ab+b^2$, equal to one for each of the three nonzero pairs $(a,b)$. Consequently this is a dimension-two binary linear matrix code, with two columns and minimum rank two. The naive Hamming formula would demand \[ 2\geq \left\lceil2\right\rceil+\left\lceil2/2\right\rceil=3, \] a contradiction. For any positive integer $t$, repeating the same matrix on $t$ diagonal blocks gives a dimension-two code of $2t$ columns and minimum rank $2t$, again violating that substitution ($2t<3t$). The puncturing bound above is attained in the $2\times2$ example: $k=(m-d+1)n=2$.

The failure has a structural explanation. A Hamming coordinate contributes an independent scalar symbol, whereas a rank-metric column here is a vector in $\mathbb F_2^2$. Treating the columns as symbols over $\mathbb F_4$ changes the logarithmic dimension and alphabet parameter. For example, the displayed two-dimensional binary space has only four codewords, not $4^2$. One cannot retain both $q=2,k=2$ and scalar Hamming coordinates while changing the metric and alphabet interpretation. A useful new Griesmer-type statement must first state these choices and its residual or puncturing operation.

Verification and adversarial audit

All four matrices of the binary example were checked by the determinant formula; block-diagonal ranks add because their column and row spaces occupy disjoint blocks. The puncturing argument uses linearity to identify the kernel and minimum rank to make it zero. The two metric parameters and the base field are never interchanged.
Audit corrections.
Distinguished binary linear dimension from dimension over a possible extension-field alphabet, and did not characterize the naive false formula as the source conjecture.
External computation.
None; symbolic arguments and boundary checks only.

Scope and remaining gaps

This supplies a known-style elementary rank bound and a counterexample to one naive transfer, not new optimal metric bounds or Solomon--Stiffler constructions for the source's broad target.
Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

Model and contributor disclosure

Codex GPT-6 completed written-proof review of this attempt at its declared scope; this is model-assisted review, not independent human proof checking. OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration and repository-format review; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.