Outline

Projective space ℙs−1 over a finite field K Definition Let K = 픽q be a finite field with q elements. The projective space of dimension s − 1 over K is the quotient space ℙs−1: = (K s ∖ {0}) / ∼ where two points 훼, 훽 in K s ∖ {0} are equivalent if 훼 = 휆훽 for some 휆 ∈ K. As usual, we denote the equivalence class of 훼 = (훼1,..., 훼s) by [훼] = [(훼1,..., 훼s)]. X ⊂ ℙs−1 and I(X) linear codes arising from X The structure of I(X) The degree of S/I(X) The geometric structure of X The case of square-free monomials Algebraic toric sets Definition Let K = 픽q be a finite field with q elements and let xvi: = xvi11 ⋅ ⋅ ⋅ xvinn, i = 1,..., s, be a finite set of monomials in R = K [x1,..., xn]. The set X: = {[(xv1,..., xvs)] ∣ xi ∈ K ∗ = K ∖ {0} ∀ i} ⊂ ℙs−1, is called an algebraic toric set parameterized by xv1,..., xvs