Expander graphs, strong blocking sets and minimal codes

Abstract

We give a new explicit construction of strong blocking sets in finite projective spaces using expander graphs and asymptotically good linear codes. Using the recently found equivalence between strong blocking sets and linear minimal codes, we give the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$ such that $n$ is at most a constant times $q k$. This solves one of the main open problems on minimal codes.

Blocking sets, minimal codes and trifferent codes

“…They proved that for every prime power $$q$$, there exists an infinite sequence of $$k$$’s such that there is an explicit construction of a strong blocking set in the projective space $$\mathrm{PG}(k - 1, q)$$ of size $$\sim q^4 k/4$$. The problem of giving an explicit construction, which also has a linear dependence on $$q$$, has recently been solved in [7]. Theorem There is an absolute constant $$c$$ such that for every prime power $$q$$ and $$k$$ large enough, there is an explicit construction of strong blocking sets in $$\mathrm{PG}(k - 1, q)$$ of size at most $$c(q + 1)k$$.…”

“…For the sake of completeness, we give a sketch of the construction in [7]. Lemma Let $$\mathcal {M}=\lbrace P_1,\ldots,P_n\rbrace$$ be a set of points in $$\mathrm{PG}(k-1,q)$$ and let $$G=(\mathcal {M},E)$$ be a graph on these points.…”

“…It is shown in [7] that constant degree expander graphs $$G$$ on $$n$$ vertices have the property $$\iota (G) = \Omega (n)$$. The explicit constructions of these graphs [43] and asymptotically good linear codes [52] thus imply Theorem 5.1, and we refer to [7] for the best values of the constant $$c$$ obtained from this construction.…”

“…The newfound equivalence between minimal codes and strong blocking sets has immensely increased the interest in proving bounds on the smallest size of a strong blocking sets and finding explicit constructions [2, 3, 7, 12, 36]. We prove the following new equivalence between strong blocking sets and certain affine 2‐blocking sets, and use it to improve the best lower and upper bounds (for $$q \geqslant 3$$) on the smallest size of a strong blocking set.…”

“…Our construction is based on a recent breakthrough on explicit constructions of strong blocking sets [7]. By restricting to $$q = 3$$, and using a different construction of strong blocking sets [12], we obtain a new explicit construction of linear trifferent codes, which improves the current best explicit construction of length‐$$n$$ linear trifferent codes from dimension $$n/112$$ (obtained in [53]) to dimension $$n/48$$.…”