A proof of the generalized packing-covering conjecture

2026-09-28

AI Summary

This study addresses the Generalized Packing Covering Conjecture, a long-standing open problem in coding theory that seeks to establish a mathematical inequality relating the generalized Hamming weights and the generalized covering radii of linear codes. We propose a novel proof framework integrating theoretical derivation with exhaustive computational verification. Specifically, higher-order cases are resolved through parity-check matrix reconstruction, length bounding, and successive shortening arguments. For lower-order cases, boundary reduction techniques transform the problem into finitely many parameter configurations, which are subsequently verified via exact enumeration algorithms. This work successfully proves that the conjecture holds for linear codes over all finite fields at any admissible order, representing a significant breakthrough in the field.

Abstract

The generalized packing--covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code $\mathcal{C}$ and every admissible order $t$, the $t$-th generalized Hamming weight $d_t(\mathcal{C})$ and the $t$-th generalized covering radius $R_t(\mathcal{C})$ satisfy $d_t(\mathcal{C})\le 2R_t(\mathcal{C})+2$. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders $t\ge 32$ and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.

Problem

• generalized packing-covering conjecture
• linear code
• generalized Hamming weight
• generalized covering radius

Innovation

• generalized packing-covering conjecture
• linear codes
• computer-assisted proof
• parity-check reformulation
• successive puncturing