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For any integral convex polytope in R 2 there is an explicit construction of an error-correcting code of length (q − 1) 2 over the finite field F q , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is… (More)

This note is meant to be an introduction to coho-mological methods and their use in the theory of error-correcting codes. In particular we consider evaluation codes on a complete intersection. The dimension of the code is determined by the Koszul complex for X ⊂ P 2 and a lower bound for the minimal distance is obtained through linkage. By way of example… (More)

We consider dynamic evaluation of algebraic functions) is an algebraic problem, we consider serving on-line requests of the form \change input x i to value v" or \what is the value of output y i ?". We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The rst gives lower bounds in a wide range of… (More)

We study subsets of Grassmann varieties G(l, m) over a field F , such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of Fq-rational points on such unions. Moreover we study a geometric duality of such unions, and give a combinatorial… (More)

Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possible altered vectorspace. Ralf Koetter and Frank R. Kschischang in Coding for errors and erasures in random network coding (IEEE Transactions on Information Theory, vol.54, no.8, pp. 3579-3591,2008) introduced a metric on the… (More)

We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersect in the same dimension. Linear network coding transmits… (More)