On the construction and labelling of geometrically uniform signal sets in ℝ2 matched to additive quotient groups

Abstract

We establish the conditions under which it is possible to construct signal sets satisfying the properties of being geometrically uniform and matched to additive quotient groups. Such signal sets consist of subsets of signal spaces identified to integers rings Z[i] and Z[ω] in R 2 .

Hyperbolic Geometrically Uniform Codes and Ungerboeck Partitioning on the Double Torus

“…Theorem 4. The Fuchsian arithmetical triangular groups which contains a subgroup of signature (2,-) are: (2, 3, 7), (2,3,8), (2,3,9), (2,3,10), (2,3,12), (2,3,18), (2,4,5), (2,4,6), (2,4,8), (2,4,12), (2,5,5), (2,5,10), (2,6,6), (2,8,8), (3,3,4), (3,3,5), (3,3,6), (3,3,9), (3,4,…”

“…It is well known that the signals of a PSK constellations of cardinality M have elements of an additive group from the ring of integers modulo as labels M, see [1]. On the other hand, the labeling of signals from a QAM constellation of cardinality M by elements of a finite group coming from a finite field appeared more directly in the works [2,3], and only by elements of an additive group G of cardinality M that need not necessarily come from a finite field, as we can see in [4].…”

“…The signal constellations QAM coming from either the Z 2 lattice or the A 2 lattice are geometrically uniform [2][3][4]. From a geometric point of view, the signals of these constellations in Z 2 and A 2 can be characterized as a finite set of points coming from a set of barycenters of squares taken from a regular tessellation by squares and by a set of barycenters of regular hexagons taken from from a hexagonal tessellation, respectively.…”

Códigos Espaço-Temporais de Treliça Baseados na Teoria de Reticulados

“…Por outro lado, ≡ 0 . Por meio da Equação (7), conclui-se que r = p r é de forma simultânea solução inteira das equações u + vr = p t e u + vr = p n . Porém, isto ocorre se, e somente se, p t = p n , ou melhor, se t = n. Voltando na equação u + p n v = p n , obtêmse como par de soluções inteiras de forma quadrática f(x, y) = x 2 + y 2 = p n .…”

“…Como consequência dos resultados obtidos em [7] via a técnica de casamento de constelações de sinais por grupos aditivos provenientes dos reticulados ℤ 2 e 2 , propôs-se um procedimento de construção de CETT com diversidade de modulação máxima, como proposto em [2], cujos resultados podem ser usados como ferramenta útil nos estudos de novos códigos.…”