Algebraic Decoding of Zetterberg and Dumer-Zinoviev Codes

We consider two families of exceptionally good double-error correcting codes: the Zetterberg binary codes and the Dumer-Zinoviev quaternary codes. The Zetterberg codes are the best known family of double-error correcting binary linear codes. They are longer than the Bose-Chaudhuri-Hocquenghem double-error correcting codes of the same redundancy. The quaternary Dumer-Zinoviev codes are the only known q-ary double-error correcting codes which asymptotically meet the Hamming bound for q >3. We derive simple criteria to decicle whether 1, 2 or 3 errors have occurred when one of these codes is used for data transmission. Based on these criteria new decoding algorithms are proposed, which are faster and simpler to implement than the known ones. The main improvements compared with the known algorithms are two. First, a quadratic equation only has to be solved when two errors have occurred. Secondly, some calculations, especially the inversion, can be carried out in a field considerably smaller than the ground field.