On the redundancy of lossy source coding

The redundancy of a source code is the difference between its expected performance and the optimum performance theoretically attainable (OPTA). In an analysis of the redundancy problem of source coding the aim is to investigate the trade-off between the minimum redundancy over a class of codes having a common parameter (such as block length) and the common parameter. We assume that the common parameter associated with the codes considered is block length. We refer to the minimum redundancy over the class of all codes having block length n and some specified type as the nth-order redundancy. In lossless source coding, the OPTA is the Shannon entropy and there exists extensive literature studying the nth-order redundancy. Before our work, nontrivial lower bounds are still unknown to either the n-th order rate redundancy or the n-th order distortion redundancy. The aim of this paper is to answer the question of whether Pilc´s (1968) lower bound is true. We derive a closed formula for the nth-order distortion redundancy and prove the formula for the upper and lower bounds of the nth-order rate redundancy