Packet Oriented Error Correcting Codes Using Vandermonde Matrices and Shift Operators

This paper proposes a design of packet oriented systematic block codes based on the Vandermonde matrix applied to a group of k information packets to construct r redundant packets which are transmitted together with the original packets into the network. The investigated codes are capable of correcting a single erroneous packet in a group of n = k+r received packets, irrespective of the number of bits in error within this packet. The elements of the Vandermonde matrix are bit level right arithmetic shift operators and this, combined with low-overhead packet padding, enables simple endcoding/decoding procedures as compared to some of the more traditional packet-level error correction approaches. The latter, similar to the codes proposed in this paper, is applicable to packets of any size with the same lengths within the block of k information packets. The correction of erroneous packet is based on syndrome decoding that provides both the location of the packet in error and the locations of the bits in error within this packet. The general code design principles are illustrated in the paper with examples of codes of different rates but the same minimum distance of three. The design performance is tested using Monte Carlo simulations and shows good agreement with theoretical results.