A contribuition to the study of channel coding in wireless communication systems

Recently, minimum and non-minimum delay perfect codes were proposed for any channel of dimension n. Their construction appears in the literature as a subset of cyclic division algebras over Q(ξ3) only for the dimension n = 2n1, where s ∈ {0,1}, n1 is odd and the signal constellations are isomorphic to Z[ξ3]. In this work, we review the cyclic division algebra and we propose an innovative methodology to extend the construction of minimum and non-minimum delay perfect codes as a subset of cyclic division algebras overQ(ξ3), where the signal constellations are isomorphic to the hexagonalA n 2 -rotated lattice, for any channel of any dimension n such that gcd(n,3) = 1. Also, interference is usually viewed as an obstacle to communication in wireless networks, so we developed a new methodology to quantize the channel coefficients in order to realize interference alignment onto a lattice. Our channel model is the same from the compute-and-forward strategy. In this new methodology, we have described a way to find an infinite nested lattice partition chain for any dimension n= 2r−2, where r ≥ 3, and we made use of the binary cyclotomic fieldQ(ξ2r), with r≥ 3. Thus, for the complex case, we developed the generalization to obtain such infinite nested lattice partition chains and we also developed a methodology for the real case. This new methodology used to solve the problem is original and can contributes greatly to the area, that is, it can be very useful in future developments.