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Theta Invariants and New Families of MRD Codes

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Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

For an $\mathbb F_{q^m}$-linear code $C\subseteq\mathbb F_{q^m}^{n}$ and a Galois automorphism $\theta$ fixing $\mathbb F_q$, put $S_i^\theta(C)=\sum_{j=0}^{i}\theta^j(C)$, $s_i^\theta(C)=\dim_{\mathbb F_{q^m}}S_i^\theta(C)$, and $\Delta_i^\theta(C)=s_{i+1}^\theta(C)-s_i^\theta(C)$. Rank distance is defined by expanding coordinates over $\mathbb F_q$. For $n\leq m$, a $k$-dimensional maximum-rank-distance (MRD) code has minimum rank distance $n-k+1$. Gabidulin codes are obtained by evaluating linearized polynomials at $\mathbb F_q$-linearly independent field elements.

Statement recorded in the supplied source.
Can the theta-dimension and theta-increment invariants of rank-metric codes characterize families of MRD codes other than Gabidulin codes, or yield new code constructions? [S2]
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Use the dimensions of successive Galois sums to recognize or construct maximum-rank-distance codes beyond the Gabidulin family.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004105 (OWR-16775-009). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Contemporary Coding Theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/13, pp. 773--840 https://doi.org/10.4171/owr/2019/13.
  • S3 Contemporary Coding Theory, Oberwolfach Reports 13/2019, primary report PDF https://ems.press/content/serial-article-files/46792. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004105 and source code OWR-16775-009 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-365-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-365

Theta Invariants and New Families of MRD Codes

Definitions and mathematical statement

Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

For an $\mathbb F_{q^m}$-linear code $C\subseteq\mathbb F_{q^m}^{n}$ and a Galois automorphism $\theta$ fixing $\mathbb F_q$, put $S_i^\theta(C)=\sum_{j=0}^{i}\theta^j(C)$, $s_i^\theta(C)=\dim_{\mathbb F_{q^m}}S_i^\theta(C)$, and $\Delta_i^\theta(C)=s_{i+1}^\theta(C)-s_i^\theta(C)$. Rank distance is defined by expanding coordinates over $\mathbb F_q$. For $n\leq m$, a $k$-dimensional maximum-rank-distance (MRD) code has minimum rank distance $n-k+1$. Gabidulin codes are obtained by evaluating linearized polynomials at $\mathbb F_q$-linearly independent field elements.

Statement recorded in the supplied source.
Can the theta-dimension and theta-increment invariants of rank-metric codes characterize families of MRD codes other than Gabidulin codes, or yield new code constructions? [S2]
Source status.
The supplied catalogue labels this record open. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Use the dimensions of successive Galois sums to recognize or construct maximum-rank-distance codes beyond the Gabidulin family.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004105 (OWR-16775-009). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Contemporary Coding Theory $2020$, Oberwolfach Reports, DOI 10.4171/owr/2019/13, pp. 773--840 https://doi.org/10.4171/owr/2019/13.
  • S3 Contemporary Coding Theory, Oberwolfach Reports 13/2019, primary report PDF https://ems.press/content/serial-article-files/46792. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30004105 and source code OWR-16775-009 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

The dimension profile needs a distance hypothesis

The primary source already proves properties of the theta profiles and gives a Gabidulin characterization with minimum rank distance greater than one. We test whether the profile alone can replace that distance hypothesis. Let $F=\mathbb F_2[\alpha]/(\alpha^3+\alpha+1)$; the cubic has no root in $\mathbb F_2$, hence is irreducible. Take $\theta(z)=z^2$ and the two $F$-linear length-three codes \[ C=\{x:x_1+\alpha x_2+\alpha^2x_3=0\},\qquad C'=\{x:x_1+\alpha x_2=0\}. \]
Proposition: Both codes have the same theta-dimension profile $(2,3,3,\ldots)$, but $C$ has minimum rank distance 2 and is MRD, while $C'$ has minimum rank distance 1.
Proof: Each is the kernel of one nonzero linear functional, so has dimension two. Its theta image has normal obtained by squaring each original normal coordinate. That new normal is not a scalar multiple of the original: their first coordinates are both 1, forcing any scalar multiple to use scalar 1, while $\alpha^2\ne\alpha$. Thus the code and its theta image are distinct two-dimensional subspaces of $F^3$, whose sum is all of $F^3$. This proves both profiles. A nonzero rank-one vector in $F^3$ is of the form $\lambda v$ with $\lambda\ne0$ and $v\in\mathbb F_2^3\setminus\{0\}$, because its coordinates span a one-dimensional $\mathbb F_2$-space. Such a vector cannot lie in $C$: $v_1+\alpha v_2+\alpha^2v_3\ne0$ by linear independence of $1,\alpha,\alpha^2$. The codeword $(\alpha,1,0)$ lies in $C$ and has rank weight 2, so its minimum is exactly 2, equal to $n-k+1$. In contrast $e_3\in C'$ has rank weight 1, proving the second claim. ∎
For completeness, the standard nonincreasing-increment property also has a short direct proof. With $S_i=\sum_{j=0}^i\theta^j(C)$, for $i\ge1$ the map induced by $\theta$ from $S_i/S_{i-1}$ onto $S_{i+1}/S_i$ is well defined, since $\theta(S_{i-1})\subseteq S_i$, and is surjective because $S_{i+1}=C+\theta(S_i)$ and $C\subseteq S_i$. This semilinear surjection cannot increase dimension, so $\Delta_i\le\Delta_{i-1}$. These necessary profile properties do not force a desired minimum distance, as the example shows.

Verification and remaining gaps

All 64 codewords of each code were enumerated over the explicitly represented field, their binary rank weights computed, and their theta-sum dimensions checked. The failed route is declaring a candidate MRD solely from this dimension profile. The example does not refute a characterization restricted in advance to MRD codes, nor the source's theorem with distance greater than one: $C'$ violates that hypothesis. No new MRD family or classification among already-MRD families is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2019/13. Relevant passages read: 2019/13 pp. 828--830, theta dimensions/increments, distance-qualified Gabidulin characterization and other-family question.

https://arxiv.org/abs/2411.19087. Opened primary abstract page for later geometric invariants of rank-metric codes; no full classification theorem is claimed from that source.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. This additional model review is not independent human verification. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch022/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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