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arXiv · 2305.15297

Strong blocking sets and minimal codes from expander graphs

Noga Alon·Anurag Bishnoi·Shagnik Das·Alessandro Neri
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Abstract

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the $(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k )$. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every prime power $q$, for which $n = O (q k)$. This solves one of the main open problems on minimal codes.

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Noga Alon, Anurag Bishnoi, Shagnik Das, Alessandro Neri. 2023-05-24. Strong blocking sets and minimal codes from expander graphs. https://arxiv.org/abs/2305.15297

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