A counterexample to the Etzion-Silberstein conjecture

2026-08-09

AI Summary

This study addresses the Etzion–Silberstein conjecture, which posits that the Singleton-type upper bound for linear Ferrers diagram rank-metric codes is always attainable. Focusing on the binary Ferrers diagram \( E = (5,5,5,5,1,1) \) with minimum rank distance 3, the authors systematically examine all 8,382,465 kernels through equivalence class classification, rank distribution analysis, kernel orbit enumeration, and a custom exhaustive verifier. They prove that the maximum achievable code dimension is 11, contradicting the conjectured value of 12. This constitutes the first counterexample to the conjecture, demonstrating its lack of universal validity. Furthermore, the work introduces the row cone propagation identity and generalizes the construction to yield an infinite family of counterexamples for all minimum rank distances at least 3.

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 $d \geq 3$.

Problem

• Etzion-Silberstein conjecture
• Ferrers diagram
• rank-metric codes
• Singleton bound
• minimum rank distance

Innovation

• Ferrers diagram
• rank-metric codes
• Etzion-Silberstein conjecture
• MRD codes
• counterexample