Minimum distances of primitive narrow-sense BCH codes via good zero-sets

Abstract

Determining the exact minimum distances of BCH codes remains a open problem. We establish the minimum distances of several families of primitive narrow-sense BCH codes, showing that they attain their designed distances. Our approach centers on 𝔽_q-good zero-sets, which we introduce through a derivative condition on their vanishing polynomials. We show that a q-ary primitive narrow-sense BCH code of length q^m−1 and designed distance 2≤δ≤ q^m−1 has minimum distance δ if and only if there exists an 𝔽_q-good zero-set of cardinality δ+1 in the finite field 𝔽_q^m with q^m elements. To construct 𝔽_q-good zero-sets, we develop several methods based on polynomial substitutions, power maps, and shifted inverses, as well as direct constructions using polynomials of special forms. Together with suitable initial 𝔽_q-good zero-sets, including those arising from known minimum-distance results, these methods yield new good zero-sets of various cardinalities and hence families of primitive narrow-sense BCH codes whose minimum distances equal their designed distances. These families cover a broad range of designed distances, with several known minimum-distance results recovered as special cases.

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BibTeX RIS

BibTeX

@misc{zheng2026minimum,
  title = {{Minimum distances of primitive narrow-sense BCH codes via good zero-sets}},
  author = {Run Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21994},
  eprint = {2609.21994},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21994},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Minimum distances of primitive narrow-sense BCH codes via good zero-sets
AU  - Run Zheng
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.21994
UR  - https://arxiv.org/abs/2609.21994
AB  - Determining the exact minimum distances of BCH codes remains a open problem. We establish the minimum distances of several families of primitive narrow-sense BCH codes, showing that they attain their designed distances. Our approach centers on 𝔽_q-good zero-sets, which we introduce through a derivative condition on their vanishing polynomials. We show that a q-ary primitive narrow-sense BCH code of length q^m−1 and designed distance 2≤δ≤ q^m−1 has minimum distance δ if and only if there exists an 𝔽_q-good zero-set of cardinality δ+1 in the finite field 𝔽_q^m with q^m elements. To construct 𝔽_q-good zero-sets, we develop several methods based on polynomial substitutions, power maps, and shifted inverses, as well as direct constructions using polynomials of special forms. Together with suitable initial 𝔽_q-good zero-sets, including those arising from known minimum-distance results, these methods yield new good zero-sets of various cardinalities and hence families of primitive narrow-sense BCH codes whose minimum distances equal their designed distances. These families cover a broad range of designed distances, with several known minimum-distance results recovered as special cases.
ER  -

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