A lower bound of the form (2n/n+1)1/nγn-1/n n-1 is derived on the coding gain γn of the densest n-dimensional (n-D) lattice(s). The bound is obtained based on constructing an n-D lattice which consists of parallel layers. Each layer is selected as a translated version of a densest (n - 1)-D lattice. The relative positioning of the layers is adjusted to make the coding gain as large as possible. For large values of n, the bound is improved through tightening Ryškov's inequality on covering radius and minimum distance of a lattice.

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Designs, Codes and Cryptography
Department of Systems and Computer Engineering

Banihashemi, A, & Khandani, A.K. (Amir K.). (1998). An Inequality on the Coding Gain of Densest Lattice Packings in Successive Dimensions. Designs, Codes and Cryptography, 14(3), 207–212. doi:10.1023/A:1008288001941