Linear codes with prescribed projective codewords of minimum weight
views
comments
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 b 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.
SWITCHcast MediaSpace video portal by Kaltura User Guides and Technical Documentation