Forms and Linear Network Codes

We present a general theory to obtain linear network codes utilizing forms and obtain explicit families of equidimensional vector spaces, in which any pair of distinct vector spaces intersect in the same small dimension. The theory is inspired by the methods of the author utilizing the osculating spaces of Veronese varieties. Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possibly altered vector space. Ralf Koetter and Frank R. Kschischang introduced a metric on the set af vector spaces and showed that a minimal distance decoder for this metric achieves correct decoding if the dimension of the intersection of the transmitted and received vector space is sufficiently large. The vector spaces in our construction are equidistant in the above metric and the distance between any pair of vector spaces is large making them suitable for linear network coding. The parameters of the resulting linear network codes are determined. Notation. • Fq is the finite field with q elements of characteristic p. • F = Fq is an algebraic closure of Fq. • Rd = F[X0, . . . , Xn]d and Rd(Fq) = Fq[X0, . . . , Xn]d the homogenous polynomials of degree d with coefficients in F and Fq. • R = F[X0, . . . , Xn] = ⊕dRd and R(Fq) = Fq[X0, . . . , Xn] = ⊕dRd(Fq). • G(l, N) is the Grassmannian of l-dimensional F-linear subspaces of F and G(l, N)(Fq) its Fq-rational points, i.e. l-dimensional Fq-linear subspaces of Fq .