Groups of algebraic integers used for coding QAM signals

Linear block codes over Gaussian integers and Eisenstein integers were used for coding over two-dimensional signal space. A group of Gaussian integers with 2²ⁿ elements was constructed to code quadrature amplitude modulation (QAM) signals such that a differentially coherent method can be applied to demodulate the QAM signals. This paper shows that one subgroup of the multiplicative group of units in the algebraic integer ring of any quadratic number field with unique factorization, modulo the ideal (Pⁿ), can be used to obtain a QAM signal space of 2p^2n-2 points, where p is any given odd prime number. Furthermore, one subgroup of the multiplicative group of units in the quotient ring Z[ω]/(pⁿ) can be used to obtain a QAM signal space of 6p^2n-2 points; one subgroup of the multiplicative group of units in the quotient ring Z[i](pⁿ) 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[ω]/(2ⁿ) can be used to obtain a QAM signal space of 3·2^2n-2 points, where i=√-1, ω=(-1+√-3)/2=(-1+i√3)/2, p is any given odd prime number, Z[i] and Z[ω] are, respectively, the Gaussian integer ring and the Eisenstein integer ring. These multiplicative groups can also be used to construct block codes over Gaussian integers or Eisenstein integers which are able to correct some error patterns