New results and open problems in real-number codes

Signal processing techniques (such as the FFT and sampling rate conversion) can be used to locate and correct data errors, as an alternative to error control coding. These error control techniques in the real or complex fields are not subject to stringent restrictions on block length, as it is the case in Galois fields, and can be readily performed in DSPs. Furthermore, they are capable of exploring the redundancy that naturally exists in oversampled signals, rather than introducing redundant data in a different domain and in a separate channel-coding step (the finite field approach). However, the use of finite-precision real arithmetic raises the question of numerical stability. We study the conditioning of an error correction algorithm in the complex field, in an attempt to understand the effect of the location of the errors and the effect of the error amplitudes. We then consider the parallel concatenation of these codes, which seems to lead to surprisingly stable codes, and formulate a number of open problems concerning them.