Characterization of Nested Lattice Codes Through Quotient Groups Relative to Normal Fuzzy Subgroups for Channel Quantization

Abstract

Interference is in general, an undesirable obstacle to communication in wireless communication networks. Therefore, in order to realize interference alignment, this work intends to characterize classical codes through the framework of fuzzy set theory for quantizing real-valued channel coefficients. Thenceforward, from the classical framework nested real lattice codes are constructed by using construction A of nested real ideal lattices and, therefore, through normal fuzzy subgroups this work shows that the constructed quotient groups of the nested real ideal lattices relative to normal fuzzy subgroups of these nested real ideal lattices are nested vector spaces and, consequently, such nested quotient groups have the structure of a linear code, are isomorphic to the nested real lattice codes and defined by linear fuzzy codes. Through such an isomorphism and the construction of the nested quotient groups relative to normal fuzzy subgroups, it is possible to conclude that the codewords of these nested linear fuzzy codes are in a one-to-one correspondence to membership vectors. Through the constructed nested quotient groups, that is, nested linear fuzzy codes, this work supplies how to quantize real-valued channel coefficients onto these nested quotient groups in order to realize interference alignment and, consequently, it is possible to encode an arbitrary real-valued channel coefficient by a codeword from a linear fuzzy code.