A codeword of a linear code is called minimal if its support (the set of nonzero coordinates) does not contain the support of any other linearly independent codeword. For a linear code \(C\), the supports of minimal codewords of the dual code \(C^\bot\) give the access structure of a secret sharing scheme, introduced by Massey.
The authors recall some results about blocking sets and introduce a new definition of cutting blocking set: An affine (respectively projective, respectively vectorial) \(k\)-blocking set is cutting if its intersection with every \((n-k)\)-dimensional affine (respectively projective, respectively affine through the origin) subspace is not contained in any other \((n-k)\)-dimensional affine (respectively projective, respectively affine through the origin) subspace. It is shown that a set \(B\) is a \(t\)-dimensional (with \(t>n-k\)) affine cutting \(k\)-blocking set if and only if the intersection between \(B\) and every \((n-k)\)-dimensional affine subspace is not contained in any other \((n-k)\)-dimensional affine subspace.
A general construction of minimal linear codes related to blocking sets, both in affine and in projective case is provided. For every function \(f:\mathbb{F}_q^n\to\mathbb{F}_q\), the code \(C_f\) is defined as the set \(\{c(u,v)\}=\{uf(x)+v\cdot x\}\) for all \(u\in F_q\) and \(v\in\mathbb{F}_q^n\), \(x\in (\mathbb{F}_q^n)^\ast,\) where \(\cdot\) is the Euclidean inner product. The parameters of \(C_f\) are found for a nonlinear function \(f\) and its minimality is shown when the affine surface \(V(f)^\ast\) is an \(n\)-dimensional cutting vectorial \((1,n-1)\)-blocking set in \(\mathbb{A}(\mathbb{F}_n^q)\) and for every nonzero vector \(v,\) it exists \(x\) such that \(f(x)+v\cdot x=0\) and \(f(x)\) is different from 0. These results are extended to the projective space taking a nonzero vector from each one-dimensional subspace of \(\mathbb{F}_n^q.\)
The last section presents an explicit family of minimal linear codes and it is proved that infinitely many of its members do not satisfy the Ashikhmin and Barg condition \(\cfrac{w_{\max}}{w_{\min}}\leq\cfrac{q}{q-1},\) where \(w_{\max}\) and \(w_{\min}\) are the maximum and minimum nonzero weights in \(C\), respectively.