Packet loss recovery codes based on Vandermonde matrices and shift operators

An increasing number of real-time applications in packet networks uses erasure codes to cope with packet losses. Most of these codes were designed originally for bit or symbol oriented transmission. This paper introduces packet oriented block codes for the recovery of lost packets. Specifically, a family of systematic erasure codes is proposed based on the Vandermonde matrix applied to a group of k information packets to construct r redundant packets. The elements of the Vandermonde matrix are bit level right arithmetic shift operators applied to the information packets. With low-overhead packet padding, the code design is applicable to packets of any size with the same lengths within the block of k information packets. The recovery of lost packets is based on inverting a matrix corresponding to the coefficient -Vandermonde-matrix augmented by the identity matrix with the rows removed according to the sequence number of the lost packets. The general code design principles are illustrated in this paper with examples of codes of different parameters. Erasure recovery capability of the proposed codes is characterized by simple decoding procedures. The code designs are tested using Monte Carlo simulations and their performance shows good agreement with theoretical results.