Algebraic-geometric codes and asymptotic problems (Q1180622)

Asymptotic bounds of coding theory are summarized and analyzed. For an \((n,k,d)_ q\) linear code let \(R=k/n\), \(\delta=d/n\), the relations between the possible limit values of \(\delta\) and \(R\) when \(n,k\) and \(d\) tend to infinity are considered. Let \(V_ q\) be the set of \((\delta,R)\) pairs for all possible \(q\)-ary codes and \(U_ q\) the set of its limit points. Similar quantities can be described for the set of linear codes and the set of codes that can be constructed with complexity polynomial in their length. The classical upper and lower bounds are considered along with more recent ones. Certain aspects of codes from algebraic geometries are surveyed as well as bounds specific to such codes. Some relationships between the various bounds are considered with tables given for the cases of \(q=2\) and 49.