Bounds on codes from smooth toric threefolds with rank(pic(x)) = 2
Kimball, James Lee
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In 1998, J. P. Hansen introduced the construction of an error-correcting code over a finite field Fq from a convex integral polytope in R2. Given a polytope P ? R2, there is an associated toric variety XP , and Hansen used the cohomology and intersection theory of divisors on XP to determine explicit formulas for the dimension and minimum distance of the associated toric code CP . We begin by reviewing the basics of algebraic coding theory and toric varieties and discuss how these areas intertwine with discrete geometry. Our first results characterize certain polygons that generate and do not generate maximum distance separable (MDS) codes and Almost-MDS codes. In 2006, Little and Schenck gave formulas for the minimum distance of certain toric codes corresponding to smooth toric surfaces with rank(Pic(X)) = 2 and rank(Pic(X)) = 3. Additionally, they gave upper and lower bounds on the minimum distance of an arbitrary toric code CP by finding a subpolygon of P with a maximal, nontrivial Minkowski sum decomposition. Following this example, we give explicit formulas for the minimum distance of toric codes associated with two families of smooth toric threefolds with rank(Pic(X)) = 2, characterized by G. Ewald and A. Schmeinck in 1993. Lastly, we give explicit formulas for the dimension of a toric code generated from a Minkowski sum of a finite number of polytopes in R2 and R3 and a lower bound for the minimum distance.