Some simple observations on lattice codes

In this paper we are considering product distance of lattice codes. We first remark that the general bounds given for the product distance of real lattice codes in coding theory literature are not the best possible. We then consider product distance bounds of rotated ¿ⁿ and ¿[i]ⁿ lattices and show that the simple and general bound can be applied here and that this bound is better than the previously used number theoretic bound in most cases. This analysis also proves that when n grows the orthogonality of ¿ⁿ and ¿[i]ⁿ lattices does bound the achievable product distance strongly. Our simple observation proves the surprising result that when n is a power of two the existing rotated ¿[i]ⁿ lattices are already the best possible. In the last section we concentrate on the block fading channel, or what is equivalent to the slow fading parallel channel, and give few new examples of DMT achieving coding schemes and compare the s-product distance of these to one known coding scheme.