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1 modulo 6 if d = -3. These alphabets are isomorphic to the field GF(p), and both will be denoted by A. Associated to any two elements of G F (p) , there is a distance, which is called Mannheim distance between the corresponding elements in A. Four classes of codes are proposed. One class is designed to correct one Mannbeim error, another to correct errors of any Mannheim weight affecting one coordinate of a code vector, another to correct errors of Mannheim weight 1 affecting two coordinates of a code vector, and another to correct errors of arbitrary Mannbeim weight affecting two coordinates of a code vector. All codes in the present paper are constacyclic codes [2]. We present efficient syndrome decoding algorithms for each class being proposed. Finally, the Berlekamp-Massey algorithm is used when multiple Hamming errors occur.Resumo ·Neste trabalho c6digos sobre os inteiros algebricos provenientes de duas extensoes do conjunto dos numerus racionais Q isto e, Q( i) e Q ( R) sao propostos. Estes c6-digos sao projetados para a distancia de Mannbeim embora algumas propriedades com rela9ao a distancia de Hannning sao tambem apresentadas, isto e, mostramos que estes sao c6digos com a maxima distancia de sepra9li0, ou equivalentemente, sao c6digos MDS. Eficientes algoritrnos de decodifica9lio sao propostos para a decodifica9lio destes c6digos quando ate duas coordenadas da palavra-c6digo transmitida sao alteradas por erros com peso de Mannbeim arbitrano. 0 algoritrno de Berlekamp-Mussey e utilizado na corr~ao de multiples erros. 0 interesse pratico destes c6digos sob a metrica de Mannheim esta relacionado com esquemas de moduIa9lio baseado em constela96es do tipo QAM para as quais nem a metrica de Hannning nem a metrica de Lee sao apro- ALGEBRAIC NUMBER FIELDSpriadas. 2.Abstract -We propose codes over the algebraic inte-In this section we review the background material on the thegers of two quadratic extensions of Q, namely,