A counterexample to the Etzion-Silberstein conjecture

Abstract

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 $d\ge 3$.

8 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