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 con-trol techniques in the real or complex elds are not subject to stringent restrictions on block length, as it is the case in Galois elds, and can be readily performed in DSPs. Fur-thermore, they are capable of exploring the redundancy that naturally exists in oversampled signals, rather than introduc-ing redundant data in a different domain and in a separate channel-coding step (the nite eld approach). However, the use of nite-precision real arithmetic raises the question of numerical stability. We study the condition-ing of an error correction algorithm in the complex eld, in an attempt to understand the effect of the location of the er-rors and the effect of the error amplitudes. We then con-sider the parallel concatenation of these codes, which seems to lead to surprisingly stable codes, and formulate a number of open problems concerning them. 1