For every linear code C⊆F_q^n of dimension k over every finite field F_q, and every admissible order t with 1≤ t≤ k, the t-th generalized Hamming weight and the t-th generalized covering radius satisfy d_t(C)≤ 2R_t(C)+2.

Generalized packing–covering conjecture

The conjecture asks whether generalized packing radii are bounded by the corresponding generalized covering radii for linear codes. A September 2026 paper claims a universal bound implying the conjecture across all finite fields, code parameters, and admissible generalized weights.

September 2026 developments

On September 28, 2026, Gianira N. Alfarano, Giuseppe Marino, Alessandro Neri, and Rocco Trombetti reported a computer-assisted proof of $d_t(C)\le 2R_t(C)+2$ , using parity-check reformulation, puncturing, a counterexample-length bound, and exact computation. The retrieved evidence does not independently verify the implementation. Separately, Wenjun Yu and Moshe Schwartz claimed results for second-order radii, low-rate codes, and sufficiently long codes; Isaac Barouch Essayag and Aryeh Lev Zabokritskiy claimed the conjecture for redundancy at most $14$ and the bound $d_t(C)\le 2R_t(C)+1$ when $q\ge R_t(C)$ .

Current status (as of September 2026): the conjecture has a claimed computer-assisted proof, while independent verification of that proof and computation is still outstanding.

Generalized packing–covering conjecture

Primary source

Additional references

Progress summary

Solutions 0