On the Existence of Cyclic Lattice Codes (2402.18094v2)
Abstract: A coding lattice $\Lambda_c$ and a shaping lattice $\Lambda_s$ forms a nested lattice code $\mathcal{C}$ if $\Lambda_s \subseteq \Lambda_c$. Under some conditions, $\mathcal{C}$ is a finite cyclic group formed by rectangular encoding. This paper presents the conditions for the existence of such $\mathcal{C}$ and provides some designs. These designs correspond to solutions to linear Diophantine equations so that a cyclic lattice code $\mathcal C$ of arbitrary codebook size $M$ can possess group isomorphism, which is an essential property for a nested lattice code to be applied in physical layer network relaying techniques such as compute and forward.
- J. Conway and N. Sloane, “A fast encoding method for lattice codes and quantizers,” IEEE Transactions on Information Theory, vol. 29, no. 6, pp. 820–824, 1983.
- B. M. Kurkoski, “Encoding and indexing of lattice codes,” IEEE Transactions on Information Theory, vol. 64, no. 9, pp. 6320–6332, 2018.
- B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interference through structured codes,” IEEE Transactions on Information Theory, vol. 57, no. 10, pp. 6463–6486, 2011.
- G. Forney, “Coset codes. I. introduction and geometrical classification,” IEEE Transactions on Information Theory, vol. 34, no. 5, pp. 1123–1151, 1988.
- J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, “Notes on word hyperbolic groups,” in Group theory from a geometrical viewpoint, 1991.
- D. Micciancio, “Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions,” in The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings, 2002, pp. 356–365.
- C. Feng, D. Silva, and F. R. Kschischang, “An algebraic approach to physical-layer network coding,” in 2010 IEEE International Symposium on Information Theory, 2010, pp. 1017–1021.
- N. E. Tunali, Y.-C. Huang, J. J. Boutros, and K. R. Narayanan, “Lattices over Eisenstein integers for compute-and-forward,” IEEE Transactions on Information Theory, vol. 61, no. 10, pp. 5306–5321, 2015.
- Y.-C. Huang, K. R. Narayanan, and P.-C. Wang, “Adaptive compute-and-forward with lattice codes over algebraic integers,” in 2015 IEEE International Symposium on Information Theory (ISIT), 2015, pp. 566–570.
- ——, “Lattices over algebraic integers with an application to compute-and-forward,” IEEE Transactions on Information Theory, vol. 64, no. 10, pp. 6863–6877, 2018.
- M. A. Vázquez-Castro and F. Oggier, “Lattice network coding over euclidean domains,” in 2014 22nd European Signal Processing Conference (EUSIPCO), 2014, pp. 1148–1152.
- A. Sakzad, E. Viterbo, J. Boutros, and Y. Hong, “Phase precoded compute-and-forward with partial feedback,” in 2014 IEEE International Symposium on Information Theory. IEEE, 2014, pp. 2117–2121.
Sponsor
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.