Binary quadratic residue codes are nominally one-half-rate codes that are interesting because of their cyclic structure and the fact that their error-correction capabilities usually beat those of other codes of comparable rates. Decoding them efficiently has been a major challenge due to the lack of the so-called strings of known consecutive syndromes. Once these are determined, the decoding of a QR code can be performed efficiently by either the Peterson-Gorenstein-Zierler or the Berlekamp-Massey algorithm. The method of determining unknown syndromes was first presented by He et al. in the early 2000s for decoding the (47, 24, 11) quadratic residue code. In this method, an unknown syndrome, say, si , is calculated as si = fi(t1, . . . , tk) where fi is a polynomial and t1, . . . , tk are known syndromes. Determining fi for QR codes of different lengths (in general) is a research problem. In this thesis we focus on the (31, 15, 8) and (41, 20, 10) QR codes. Although decoding algorithms for them are well known in the literature, the ones proposed here will correct errors beyond the error-correction capabilities of those codes. Ultimately, the goal of having such algorithms is to improve the performance of the codes when they are used in very noisy channels.
