We consider the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal domains (PID). A novel scheme called adaptive compute-andforward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive computeand-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.
Index TermsCompute-and-forward, physical-layer network coding, lattice codes, and algebraic integers. On one hand, since Z[i] and Z[ω] are instances of imaginary quadratic integers, it seems natural to extend the compute-and-forward framework to general rings of imaginary quadratic integers. On the other hand, since the role of the underlying ring of integers can be effectively thought of as being a quantizer of the channel and Z[ω] is already the best quantizer for C where the channel coefficients belong, it seems unnecessary to pursue compute-and-forward over other rings of integers. In this paper, we first seek to better understand the role of rings of algebraic integers in constructing good lattices. We then use an example, namely compute-and-forward with limited feedback, to demonstrate the benefits of performing compute-and-forward over rings of imaginary quadratic integers other than Z[i] and Z [ω].One important difference between a general ring of imaginary quadratic integers and the Gaussian and Eisenstein integers is that the Gaussian integers and Eisenstein integers are not merely rings, they are principal ideal domains (PIDs). Hence, every ideal is generated by a singleton and one can equivalently work with numbers instead of ideals. The constructions of lattices over these two rings [3]- [5] heavily rely on properties of PID. However, a general ring of imaginary quadratic integers is not a PID. Therefore, in