In this note we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.

Let K = 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-Mullertype codes. As a consequence we recover some results on projective Reed-Muller-type codes over the Segre variety and over projective tori.

In this paper we give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields.

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