What was already known
Complete tradeoffs were known for (4,3,3) and (5,4,4), and for general k=d in the restricted linear-code class. An earlier (6,5,5) outer bound left one segment unresolved. The new conditioning reduction and exact-certificate search close that gap and give the (7,6,6) converse.
AI involvement
Tian reports that GPT-6 Astra proposed the conditional shortened codes and conducted the restricted entropy-constraint search, then produced the submodularity-based argument, certificates and initial verifier/draft. Tian supplied prior references and an induction suggestion, checked reasoning steps, reviewed/tested the script and revised the paper.
Importance
Extends exact information-theoretic converse results to two specified finite code models, including codes that are not restricted to linear encoding.
Limitations
The result does not characterize arbitrary n,k,d or the (8,7,7) case. Layered-code constructions are prior human work. The ten exact integer certificates and verifier are described in the paper, but both linked artifact hosts were unavailable during this review; complete certificate execution remains unconfirmed.
Technical verification
Chao Tian reports verifying the reasoning steps and reviewing/testing the Python verifier. This supports partial checking. The linked repository and companion page returned 404 during review, so execution of the complete ten-certificate package was not established.
Independent checking
Chao Tian reports verifying the reasoning steps and reviewing/testing the Python verifier. This supports partial checking. The linked repository and companion page returned 404 during review, so execution of the complete ten-certificate package was not established.
- Checker
- Chao Tian (author)
- Scope
- Reported human checking of reasoning steps and script review/tests; not a reproduced execution of all certificates.
- Basis
- Reported by the linked checker
Human understanding
Tian explains the conditioning/shortening mechanism and how the restricted entropy LP yields the converse, reports checking its reasoning, and connects it to prior code bounds. Difficulty directly reading the large integer certificates concerns verification, not the central mechanism.
Correction and publication history
Added the sourced contribution with its stated hypotheses, AI role, prior work and checking limits.
Read this published revision