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

The lattice $mathcal{L}(A)$ of a full-column rank matrix $Ain mathbb{R}^{mtimes n}$ is defined as the set of all the integer linear combinations of the column vectors of $A$ The successive minima $lambda_i(A),,1leq ileq n,$ of lattice $mathcal{L}(A)$ are important quantities since they have close relationships with the following problems: 1) shortest vector problem, 2) shortest independent vector problem; 3) successive minima problem. These problems arise from many applications, such as communications and cryptography. This paper first investigates some properties of $lambda_i(A)$. We develop lower and upper bounds on $lambda_i(A)$, where $A$ are respectively the Cholesky factor of $G_1+G_2$ and $(G_1+G_2)^{-1}$ for two given symmetric positive definitive matrices $G_1$ and $G_2$. Then we show how some properties of $lambda_i(A)$ are used to design a suboptimal integer-forcing strategy for cloud radio access network. Our approach provides higher efficiency, while keeping the same achievable rate as the algorithm reported by Bakoury et al. Simulation tests are performed to illustrate our main results.