  Universal security over a network with linear network coding has been
intensively studied. However, previous linear codes used for this purpose were
linear over a larger field than that used on the network. In this work, we
introduce new parameters (relative dimension/rank support profile and relative
generalized matrix weights) for linear codes that are linear over the field
used in the network, measuring the universal security performance of these
codes. The proposed new parameters enable us to use optimally universal secure
linear codes on noiseless networks for all possible parameters, as opposed to
previous works, and also enable us to add universal security to the recently
proposed list-decodable rank-metric codes by Guruswami et al. We give several
properties of the new parameters: monotonicity, Singleton-type lower and upper
bounds, a duality theorem, and definitions and characterizations of
equivalences of linear codes. Finally, we show that our parameters strictly
extend relative dimension/length profile and relative generalized Hamming
weights, respectively, and relative dimension/intersection profile and relative
generalized rank weights, respectively. Moreover, we show that generalized
matrix weights are larger than Delsarte generalized weights.
