Novel Properties of Successive Minima and Their Applications to 5G Tactile Internet

Abstract

The lattice $\mathcal {L}(\boldsymbol{A})$ of a full-column rank matrix $\boldsymbol{A}\in \mathbb {R}^{m\times n}$ is defined as the set of all the integer linear combinations of the column vectors of $\boldsymbol{A}$ . The successive minima $\lambda _i(\boldsymbol{A}),\,1\leq i\leq n,$ of lattice $\mathcal {L}(\boldsymbol{A})$ are important quantities since they have close relationships with the following problems: shortest vector problem, shortest independent vector problem, and successive minima problem. These problems arise from many practical applications, such as communications and cryptography. This paper first investigates some properties of $\lambda _i(\boldsymbol{A})$ . Specifically, we develop lower and upper bounds on $\lambda _i(\boldsymbol{A})$ , where $\boldsymbol{A}$ are, respectively, the Cholesky factor of $\boldsymbol{G}_1+\boldsymbol{G}_2$ and $(\boldsymbol{G}_1+\boldsymbol{G}_2)^{-1}$ for two given symmetric positive definitive matrices $\boldsymbol{G}_1$ and $\boldsymbol{G}_2$ . The bounds are, respectively, expressed as the successive minima of $\mathcal {L}(\boldsymbol{A}_1)$ and $\mathcal {L}(\boldsymbol{A}_2)$ , and $\mathcal {L}(\hat{\boldsymbol{A}}_1)$ and $\mathcal {L}(\hat{\boldsymbol{A}}_2)$ , where $\boldsymbol{A}_1, \boldsymbol{A}_2, \hat{\boldsymbol{A}}_1$ and $\hat{\boldsymbol{A}}_2$ are, respectively, the Cholesky factors of $\boldsymbol{G}_1, \boldsymbol{G}_2, \boldsymbol{G}_1^{-1}$ , and $\boldsymbol{G}_2^{-1}$ . Then, we show how some properties of $\lambda _i(\boldsymbol{A})$ are used to design a suboptimal integer-forcing strategy for cloud radio access network. Our approach provides much higher time efficiency while keeping the same achievable rate as the algorithm reported by Bakoury and Nazer (I. E. Bakoury and B. Nazer, “Integer-forcing architectures for uplink cloud radio access networks,” in Proc. 55th Annu. Allerton Conf. Commun. Control Comput. , Oct. 2007, pp. 67–75). Simulation tests are performed to illustrate our main results.