Determination of the Number of Errors in DFT Codes Subject to Low-Level Quantization Noise

This paper analyzes the effects of quantization or other low-level noise on the error correcting capability of a popular class of real-number Bose-Chaudhuri-Hocquenghem (BCH) codes known as discrete Fourier transform (DFT) codes. In the absence of low-level noise, a modified version of the Peterson-Gorenstein-Zierler (PGZ) algorithm allows the correction of up to corrupted entries in the real-valued code vector of an DFT code. In this paper, we analyze the performance of this modified PGZ algorithm in the presence of low-level (quantization or other) noise that might affect each entry of the code vector (and not simply of them). We focus on the part of the algorithm that determines the number of errors that have corrupted the real-number codeword. Our approach for determining the number of errors is more effective than existing systematic approaches in the literature and results in an explicit lower bound on the precision needed to guarantee the correct determination of the number of errors; our simulations suggest that this bound can be tight. Finally, we prove that the optimal bit allocation for DFT codes (in terms of correctly determining the number of errors) is the uniform one.