Twisted group codes of dimension 2 and 3

Source abstract

It is known that all group codes of dimension at most $3$ are permutation equivalent to abelian group codes. We prove that twisted group codes of dimension at most $3$ are monomially equivalent either to an abelian group code or to an irreducible constacyclic code, and in the second case we describe the code explicitly by means of a trace map. Monomial equivalence cannot be replaced by permutation equivalence, and the second case cannot be avoided: the ternary tetracode is an ideal of $\mathbb F_3[x]/(x^4+1)$ but it is not monomially equivalent to any group code. More generally, we give two infinite families, in dimensions $2$ and $3$, of irreducible constacyclic codes which attain the Griesmer bound, whereas every group code with the same length and dimension has smaller minimum distance.