Generalizing Linear Real-Field Codes for Fading Channels

| We derive generalized analytical constructions of linear realeld (LRF) codes for transmissions over wireless fading channels. We show cases where LRF-coded QAM, PAM, or PSK constellations can achieve maximum diversity and large coding gains. We construct analytically LRF codes, which not only yield larger coding gains than existing designs in most cases, but also produce LRF-coded QAM or PAM with desirable constellation characteristics. We also disclose an inherent connection between the optimality of LRF code construction over QAM or PAM, and a long-held mathematical conjecture in the theory of geometry of numbers. In certain cases, these results allow us to conjecture the optimality of our LRF codes over these constellations. Simulations corroborate our theoretical ndings.