Some constructions of linearly optimal group codes

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]q-codes over Fq for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed–Solomon codes as ideals of the group ring FqG where G is an elementary abelian group of order q.