Coding to reduce delay on a permutation channel

This paper introduces and analyzes the permutation channel: an error free channel that takes a block of M codewords as the input, and returns a random permutation of the codewords as output, where the permutation reflects the sequence of time instants at which the codewords are received. The decoding delay of a dataword is defined to be proportional to the earliest time that the dataword and all preceding datawords are recoverable from the currently held codewords. The problem is motivated by practical networking scenarios where packet reordering at the receiver may limit the performance of an application, e.g., a media stream where playback may be stalled on account of waiting for a particular frame to arrive. The key performance tradeoff is the code rate and decoding delay. Intuitively decoding delay may be reduced by spreading datawords across multiple codewords, but this redundancy lowers the code rate. We introduce a zero delay code, also known as a priority encoded transmission code, based on independent linear combinations of portions of the datawords. We generalize the latter code to non-zero delays, and formulate the rate delay tradeoff problem for large block lengths as a calculus of variations problem, which we then solve. We establish that our code is able to achieve an asymptotic code rate of (1− log √ δ)−1 when subject to a delay bound of δ. Finally, we note that the permutation channel is a particular instance of the degraded broadcast channel, where receivers correspond to information available at the codeword reception instants, establishing the rate optimality of the codes introduced.