Blocking sets, minimal codes and trifferent codes
share
We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-2 subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least 3. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set. Over the finite field 𝔽_3, we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length n has size at most 3^n/4.55, improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length n and size at least 1/3( 9/5 )^n/4, thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension-2 subspaces that are a constant factor bigger than the best known lower bound. By restricting to 𝔽_3, we obtain linear trifferent codes of size at least 3^7n/240, improving the current best explicit construction that has size 3^n/112.
READ FULL TEXT