Exact minimum distances of primitive narrow-sense BCH codes

Determine the exact minimum distances of primitive narrow-sense BCH codes of length q^m-1 for general designed distances, beyond the families for which equality with the designed distance is established.

Background

Primitive narrow-sense BCH codes have minimum distance at least as large as their designed distance by the BCH bound, but equality is known only for selected parameter families. The paper develops F_q-good zero-sets as a criterion for proving equality in additional families, while the general determination of exact minimum distances remains unresolved.

The problem concerns a broad class of q-ary BCH codes C_(q,m,delta) of length qm-1, with designed distance 2leqdeltaleqqm-1. Solving it would also clarify when the minimum distance equals the designed distance or the larger Bose distance.

References

Determining the exact minimum distances of BCH codes remains a open problem.

— Minimum distances of primitive narrow-sense BCH codes via good zero-sets  (2609.21994 - Zheng, 18 Sep 2026) in Abstract; Introduction, Section 1

These results verify two additional cases of Conjecture~2, which predicts equality of the minimum and Bose distances of $\mathcal{C}_{(q,m,qt+1)}$.

— Minimum distances of primitive narrow-sense BCH codes via good zero-sets  (2609.21994 - Zheng, 18 Sep 2026) in Introduction, paragraph discussing Theorem 9 and Conjecture 2