Grassmannian Codes with New Distance Measures for Network Coding

Abstract

Subspace codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. In both cases, the subspace distance is used as the distance measure. In this work we show that we can replace the subspace distance with two other possible distance measures which generalize the subspace distance. We prove that each code with the largest number of codewords and the generalized distance, given the other parameters, has the minimum requirements needed to solve a given multicast network with a scalar linear code. We discuss lower and upper bounds on the sizes of the related codes.