On groups of complex integers used as QAM signals (Q1818957)

The authors use groups of complex integers to construct two-dimensional signal constellations. In fact, they consider \(\mathbb{Z}[i]= \{a+ bi\mid a\in \mathbb{Z}\), \(b\in \mathbb{Z}\}\). They show that for \(n\geq 3\), the multiplicative group of units in the quotient ring \(\mathbb{Z}[i]/ 2^n\) is the direct product of three cyclic groups with orders \(2^{n-1}\), \(2^{n-2}\), and \(2^2\), respectively. Using the same ideas as by \textit{J. Rifà} [IEEE Trans. Inf. Theory 41, No. 5, 1512-1517 (1995; Zbl 0831.94007)], the authors construct one error-correcting code. The required modulo operation, however, is simpler, and the set of error values (after quantization) that can be corrected is larger.