Linear codes with prescribed projective codewords of minimum weight

Consider $C$, an $[n,k,d]$-linear code. Every projective codeword of minimum weight $d$ corresponds to a point in $\mathbb P^{k-1}$, and there are strong connections between the algebraic and geometric properties of these points and the parameters of $C$, especially with the minimum distance $d$. The most non-trivial connection is the fact that the Castelnuovo-Mumford regularity of the coordinate ring of these points is a lower bound for $d$. Conversely, given a finite set of points $X$ in $\mathbb P^{k-1}$, it is possible to construct linear codes with projective codewords of minimum weight corresponding to $X$. We will discuss about these constructions, and we will also look at the particular case when the constructed linear code has minimum distance equal to the regularity.