Quasi-twisted codes as contractions of quasi-cyclic codes

We consider the quasi-twisted codes as contractions of quasi-cyclic codes and construct a family of $ q $-ary quasi-cyclic codes whose codewords have $ r $-divisible weights, where $ r\mid q-1 $. We show that any quasi-cyclic code of co-index divisible by $ r $ is a direct sum of $ r $-divisible quasi-cyclic codes.