Description
Among the binary linear codes of length 47 and dimension 23, the quadratic residue code is the one that offers the highest error-correction capability: its minimum distance is equal to 12; therefore, it corrects any combination of five or fewer errors in any block of length 47. In this work, using the technique of determining unknown syndromes, we propose an algorithm for decoding that code beyond its error-correction capability. This is our main contribution. More specifically, our method determines enough unknown syndromes from the received word so that all error patterns of weight up to five and 80% of the correctable error patterns of weight six can be corrected by either the Peterson-Gorenstein-Zierler or Berlekamp-Massey algorithm. As an introduction to the technique being proposed, we illustrate its use with the quadratic residue code of length 31 and dimension 15. This code has minimum distance equal to eight and our method, also new, allows the code to correct all error patterns of weight up to three and 80% of the correctable error patterns of weight four.
