Recent advances in DFT codes based quantized frame expansions for erasure channels

Abstract

Overcomplete frame expansions have been introduced recently as signal representations resilient to erasures on packet networks such as the Internet and wireless networks. This paper is dedicated to a broad analysis of a class of frames derived from the generator matrices of DFT codes. In contrast to earlier presentations, which focused on the analysis of only low-pass DFT codes, this analysis aims at comparing the performance of all BCH codes and MDS negacyclic codes derived from the DFT codes. It is proven that all BCH codes, and MDS negacyclic codes derived from the BCH codes, are equivalent in the sense that they produce the same mean square error averaged over all erasure patterns of a given length under the same quantization error model. On the way to this result, the special property of the frames associated with the BCH codes, and the reconstruction equivalence between the frame-theoretic approach and the coding-theoretic approach, are proved in general settings. The paper also discusses optimal packetization of BCH codevectors and the performance of non-BCH codes that have the same erasure recovery capacity as BCH codes.