GROUPS OF ALGEBRAIC-INTEGERS USED FOR CODING QAM SIGNALS

Recently, linear block codes over Gaussian integers and Eisenstein int egers were used for coding over two-dimensional signal space. Group of Gaussian integers with 2(2n) elements was constructed to code quadrat ure amplitude modulation (QAM) signals such that a differentially cohe rent method can be applied to demodulate the QAM signals. This paper s hows that one subgroup of the multiplicative group of units in the alg ebraic integer ring of any quadratic number field with unique factoriz ation, module the ideal (p(n)), can be used to obtain a QAM signal spa ce of 2p(2n-2) points, where p is any given odd prime number. Furtherm ore, one subgroup of the multiplicative group of units in the quotient ring Z[omega]/(p(n)) can be used to obtain a QAM signal space of 6p(2 n-2) points; one subgroup of the multiplicative group of units in the quotient ring Z[i]/(p(n)) can be used to obtain a QAM signal space of 4p(2n-2) points which is symmetrical over the quadrants of the complex plane and useful for differentially coherent detection of QAM signals ; the multiplicative group of units in the quotient ring Z[omega]/(2(n )) can be used to obtain a QAM signal space of 3 . 2(2n-2) points, whe re i = root-1, omega = (-1 + root-3)/2 = (-1 + i root 3)/2, p is any g iven odd prime number, Z[i] and Z[omega] are, respectively, the Gaussi an integer ring and the Eisenstein integer ring. These multiplicative groups can also be used to construct block codes over Gaussian integer s or Eisenstein integers which are able to correct some error patterns .