On optimal linear codes over \(\mathbb F_5\)
The authors solve some open cases of the main coding theory problem (which consists of finding the smallest value \(n_q(k,d)\) of \(n\) for which there exists a linear code of length \(n\) and fixed dimension \(k\) and minimal distance \(d\) over GF\((q)\), the finite field with \(q\) elements). In a table, lower and upper bounds for \(n_5(5,d)\) with \(d\leq 5^4\) are summarized (including some existence and non-existence results proved by the authors or obtained by computer search and including the results of an online table of M.\ Grassl). The table shows that many cases are still open. The geometric method used to construct or to prove the non-existence of some linear codes is to consider the points of a certain subset or multiset of a projective space over GF(5) as the columns of a generator matrix of the code; here the authors use as such subsets so-called minihypers constructed, for instance, from \(n\)-arcs or blocking sets.
scientific article; zbMATH DE number 5565667
scientific article