A Simple Explicit Construction of Smaller Strong Blocking Sets
Abstract
Alon, Bishnoi, Das and Neri constructed sequences of strong blocking sets in (k−1)-dimension projective space P(bf Fq, k−1) with sizes at most Cqk, where k goes to the infinity and C is a fixed positive constant not depending on q, in their paper published in Transactions AMS, vol. 277, pp. 5389-5410, 2024. This is equivalent to a sequence of minimal [ni,ki]q codes with positive rate ki/ni ≥ c, where ki goes to the infinity and c is a fixed positive constant only depending on q. They solved one of the main open problems on minimal codes. The Alon-Bishnoi-Das-Neri construction was based on asymptotically good sequences of algebraic geometric codes and expander graphs. However, the constant C is large for small prime power q. In this paper, based on a simple approach using (q−1)-power intersecting algebraic geometric codes via concatenation to construct minimal codes, we construct explicit strong blocking sets in (k−1)-dimension projective space with sizes around 2(q2−1)k, equivalently, explicit minimal [n,k]q codes with length around 2(q2−1)k. This is an essential improvement on the previous result of Alon-Bishnoi-Das-Neri, when q ≤ 11. For large q, strong blocking sets and minimal codes in our construction are worse than the previous construction of Alon-Bishnoi-Das-Neri. However, our approach is much simpler and does not depend on expander graphs.
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BibTeX
@misc{chen2026simple,
title = {{A Simple Explicit Construction of Smaller Strong Blocking Sets}},
author = {Hao Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12819},
eprint = {2610.12819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12819},
note = {CC0},
}
RIS
TY - UNPB
TI - A Simple Explicit Construction of Smaller Strong Blocking Sets
AU - Hao Chen
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12819
UR - https://arxiv.org/abs/2610.12819
AB - Alon, Bishnoi, Das and Neri constructed sequences of strong blocking sets in (k−1)-dimension projective space P(bf F_q, k−1) with sizes at most Cqk, where k goes to the infinity and C is a fixed positive constant not depending on q, in their paper published in Transactions AMS, vol. 277, pp. 5389-5410, 2024. This is equivalent to a sequence of minimal [n_i,k_i]_q codes with positive rate k_i/n_i ≥ c, where k_i goes to the infinity and c is a fixed positive constant only depending on q. They solved one of the main open problems on minimal codes. The Alon-Bishnoi-Das-Neri construction was based on asymptotically good sequences of algebraic geometric codes and expander graphs. However, the constant C is large for small prime power q. In this paper, based on a simple approach using (q−1)-power intersecting algebraic geometric codes via concatenation to construct minimal codes, we construct explicit strong blocking sets in (k−1)-dimension projective space with sizes around 2(q^2−1)k, equivalently, explicit minimal [n,k]_q codes with length around 2(q^2−1)k. This is an essential improvement on the previous result of Alon-Bishnoi-Das-Neri, when q ≤ 11. For large q, strong blocking sets and minimal codes in our construction are worse than the previous construction of Alon-Bishnoi-Das-Neri. However, our approach is much simpler and does not depend on expander graphs.
ER -