A sharper lower bound on Rankin's constant

Rankin's constant is an important lattice constant which has applications in many fields including cryptography and communications. In spite of its importance, few of exact values of Rankin's constant are known. In this paper, we develop a lower bound on Rankin's constant ${\gamma }_{2k,k}$ which corresponds to the half volume problem. Compared with the previous best lower bound developed by Wen et al., ours is more than $\frac{\sqrt{k}}{2}$ times better. This improved lower bound on Rankin's constant directly leads to a sharper lower bound on Schnorr's constant and helps to better understand the intrinsic limitations of the 2k-block-Rankin reduction.