Unifying notions of generalized weights for security on wire-tap networks and other applications
Talk by Dr. Umberto Martinez-Penas
Date: 26.10.16 Time: 15.00 - 16.00 Room:
In this work, we propose a new kind of generalized weights, called “relative generalized matrix weights” (RGMWs), that measure error correction capability together with information leakage for any code that is linear over the field used on a network where linear network coding is applied. As applications in network coding, we obtain: 1) optimally secure linear codes for noiseless networks and all parameters, as opposed to previous works, and 2) we add universal security to the first list-decodable rank-metric codes with lists of polynomial size on their lengths, which has been recently proposed by Guruswami et al.On the theoretical side, RGMWs extend both relative generalized rank weights and relative generalized Hamming weights, and generalized matrix weights (GMWs) are larger (strictly in some cases) than Delsarte generalized weights.
We also show that the family of Galois closed spaces coincide with that of rank support spaces, and in addition, we show that both coincide with the family of left modules on the ring of matrices $F_q^(m×m)$ (m corresponding to the packet length used in the network). This also provides a connection with the recent rank error-correcting pairs and allows to treat rank weights by using products of matrices, which suggests ways of translating results in the Hamming metric that make use of the coordinate-wise product to the rank metric, such as the Feng-Rao bounds on the minimum distance or generalized weights.
Parts of this work have been presented at the 54th Annual Allerton Conference on Communication, Control, and Computing, and is a joint work with Ryutaroh Matsumoto (Tokyo Institute of Technology).