Strong blocking sets and minimal codes from expander graphs
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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 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 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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