We study improved degree distribution for Luby Transform (LT) codes which exhibits improved bit error rate performance particularly in low overhead regions. We construct the degree distribution by modifying Robust Soliton distribution. The performance of our proposed LT codes is evaluated and compared to the conventional LT codes via And-Or tree analysis. Then we propose a transmission scheme based on the proposed degree distribution to improve its frame error rate in full recovery regions. Furthermore, the improved degree distribution is applied to distributed multi-source relay networks and unequal error protection. It is shown that our schemes achieve better performance and reduced complexity especially in low overhead regions, compared with conventional schemes.