Tensor Product DFT Codes vs Standard DFT Codes

Abstract

We present a new class of linear error-correcting codes taking numerical data into codewords with numerical symbols. These codes can correct large random numerical errors added to codewords. The goal is for numerical data to be protected directly as numbers whether in storage or transmission. The coding structures are based on discrete Fourier transform (DFT) codes, defined by parity-check matrices where rows are consecutively indexed DFT vectors. The new tensor codes use the Kronecker product of two parity-check DFT matrices of shorter length codes. We construct all necessary processing matrices. Error correction methods involve two levels of syndromes and use several stochastic Berlekamp-Massey Algorithms (sBMA) to find big syndromes, locate and evaluate large errors. Decoding is done in two stages with stage one producing corrected syndromes. Stage two determines errors within a well-defined segment for the codewords. Tensor codes have simpler and more efficient decoding operations. The characteristics of the tensor codes are compared to standard DFT codes of equal length. The processing operations are significantly less for the tensor codes but the standard DFT codes sometimes have better correcting performance. Nevertheless, tensor product codes can have acceptable levels of correction, more efficiently.