AI 中文总结
该研究构造了Etzion-Silberstein猜想的反例，证明支撑于特定费勒斯图的二元线性秩距离码无法达到Singleton型上界，通过核枚举等方法验证结果，还得到行锥传播恒等式可生成系列反例。
AI 中文摘要
Etzion-Silberstein猜想断言，线性费勒斯图秩距离码的Singleton型上界对任意费勒斯图、最小秩距离和有限域均可达到。设E为列高度为(5,5,5,5,1,1)的费勒斯图，其最小秩距离为3时的界为12。我们证明，所有支撑在E上、最小秩距离为3的二元线性码的维数至多为11，并给出了一个维数为11的显式码，因此最优值恰好为11，该猜想不成立。不存在性证明将假设的维数12的码归约为二元[4×4,12,2]MRD码的三个等价类之一。秩分布论证排除了两个类，留下域类中的四个核轨道；所有四个精确提升系统均不可满足。独立编写的验证程序重现了该结果，包括对全部8,382,465个核的无轨道归约的原始枚举。我们还证明了一个精确的行锥传播恒等式，迭代该恒等式可在所有最小秩距离d≥3时产生界为12、最优值为11的二元反例。
英文摘要
The Etzion-Silberstein conjecture asserts that the Singleton-type upper bound for linear Ferrers-diagram rank-metric codes is attained for every Ferrers diagram, minimum rank distance, and finite field. Let $E$ be the Ferrers diagram with column heights $(5,5,5,5,1,1)$. The bound for minimum rank distance $3$ is $12$. We prove that every binary linear code supported on $E$ with minimum rank distance $3$ has dimension at most $11$, and we give an explicit code of dimension $11$. Thus the optimum is exactly $11$, disproving the conjecture. The nonexistence proof reduces a hypothetical dimension-$12$ code to one of the three equivalence classes of binary $[4\times 4,12,2]$ MRD codes. A rank-distribution argument eliminates two classes and leaves four kernel orbits in the field class; all four exact lift systems are unsatisfiable. Independently written verifiers reproduce the result, including a raw enumeration of all $8,382,465$ kernels without orbit reduction. We also prove an exact row-cone propagation identity. Iterating it produces binary counterexamples with bound $12$ and optimum $11$ at every minimum rank distance $d \geq 3$.
Comments8 pages. Ancillary files include the explicit certificates for E_4, E_5, and the dimension-11 code, four independent verifiers, and sample DIMACS instances; the complete verification package (all DRAT proofs, three solvers, raw enumeration, CI-attested artifacts) is at https://github.com/infinityscroll/etzion-silberstein-counterexample
