We apply algebraic geometry to two problems in error-correcting codes. We construct error-correcting codes in two ways from nonsingular projective varieties and discuss the possible duality between them. The problem of classifying space-time codes leads one to the problem of classifying non-associative division algebras over a finite field. This leads one to investigate homogeneous polynomials of degree n in n variables over Fq with no nontrivial Fq -rational points. The case n = 3 was first investigated by Dickson, and for n = 4 we give a complete geometrical classification of all possible forms.
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