Optimal linear transmission by loss-insensitive packet encoding

The objective of this paper is to characterize the optimal use of redundancy in transmitting a signal that is encoded in terms of packets of linear coefficients. The signals considered here are vectors in a finite-dimensional real or complex Hilbert space. For the purpose of transmission, these vectors are encoded in a set of linear coefficients that is partitioned in packets of equal size. We investigate how the encoding performance depends on the degree of redundancy it incorporates and on the amount of data-loss when packets are either transmitted perfectly or lost in their entirety. The encoding performance is evaluated in terms of the maximal Euclidean norm of the reconstruction error occurring for the transmission of unit vectors. Our main result is the derivation of error bounds as well as the characterization of optimal encoding when up to three packets are lost.