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

The lattice <inline-formula><tex-math notation="LaTeX">$\mathcal {L}(\boldsymbol{A})$</tex-math></inline-formula> of a full-column rank matrix <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}\in \mathbb {R}^{m\times n}$</tex-math></inline-formula> is defined as the set of all the integer linear combinations of the column vectors of <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}$</tex-math></inline-formula>. The successive minima <inline-formula><tex-math notation="LaTeX">$\lambda _i(\boldsymbol{A}),\,1\leq i\leq n,$</tex-math></inline-formula> of lattice <inline-formula><tex-math notation="LaTeX">$\mathcal {L}(\boldsymbol{A})$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\lambda _i(\boldsymbol{A})$</tex-math></inline-formula>. Specifically, we develop lower and upper bounds on <inline-formula><tex-math notation="LaTeX">$\lambda _i(\boldsymbol{A})$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}$</tex-math></inline-formula> are, respectively, the Cholesky factor of <inline-formula><tex-math notation="LaTeX">$\boldsymbol{G}_1+\boldsymbol{G}_2$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$(\boldsymbol{G}_1+\boldsymbol{G}_2)^{-1}$</tex-math></inline-formula> for two given symmetric positive definitive matrices <inline-formula><tex-math notation="LaTeX">$\boldsymbol{G}_1$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{G}_2$</tex-math></inline-formula>. The bounds are, respectively, expressed as the successive minima of <inline-formula><tex-math notation="LaTeX">$\mathcal {L}(\boldsymbol{A}_1)$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal {L}(\boldsymbol{A}_2)$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$\mathcal {L}(\hat{\boldsymbol{A}}_1)$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal {L}(\hat{\boldsymbol{A}}_2)$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}_1, \boldsymbol{A}_2, \hat{\boldsymbol{A}}_1$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\hat{\boldsymbol{A}}_2$</tex-math></inline-formula> are, respectively, the Cholesky factors of <inline-formula><tex-math notation="LaTeX">$\boldsymbol{G}_1, \boldsymbol{G}_2, \boldsymbol{G}_1^{-1}$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{G}_2^{-1}$</tex-math></inline-formula>. Then, we show how some properties of <inline-formula><tex-math notation="LaTeX">$\lambda _i(\boldsymbol{A})$</tex-math></inline-formula> 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 <italic>Proc. 55th Annu. Allerton Conf. Commun. Control Comput.</italic>, Oct. 2007, pp. 67–75). Simulation tests are performed to illustrate our main results.