Lattice constellations and codes from quadratic number fields - Pires Da Nóbrega Neto, T. [UNESP]; Interlando, J. C. [UNESP]; Favareto, O. M. [UNESP]; Elia, M. [UNESP]; Palazzo R., Jr [UNESP] (Institute of Electrical and Electronics Engineers (IEEE), 2001)
We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric modulo a two-dimensional (2-D) grid. In particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.
Keywords: Algebraic decoding; Euclidean domains; Lattices; Linear codes; Mannheim distance; Number fields; Signal sets matched to groups; Algorithms; Codes (symbols); Decoding; Error analysis; Linearization; Maximum likelihood estimation; Maximum principle; Number theory; Quadratic programming; Quadrature amplitude modulation; Two dimensional; Vector quantization; Einstein-Jacobi integers; Gaussian integers; Hamming distance; Lattice codes; Lattice constellations; Manhattan metric modulo; Mannheim metric; Maximum distance separable; Quadratic number fields; Information theory