Direct products in projective Segre codes

Let K=FqK=Fq be a finite field. We introduce a family of projective Reed–Muller-type codes called projective Segre codes. Using commutative algebra and linear algebra methods, we study their basic parameters and show that they are direct products of projective Reed–Muller-type codes. As a consequence we recover some results on projective Reed–Muller-type codes over the Segre variety and over projective tori.