An exact solution to degree distribution optimization in LT codes

Since their invention Luby Transform (LT) codes have been regarded as an efficient capacity achieving channel code over binary erasure channels. Majority of the previous research is focused on designing the degree distribution of LT codes which are asymptotically optimal. Nevertheless, deigning the optimal degree distribution of LT codes for finite and small message lengths remains as an open problem. This work addresses the degree distribution optimization of LT codes over binary erasure channel. Based on AND-OR tree analysis of Belief Propagation (BP) algorithm, a new formulation of the problem in the form of standard semidefinite programming is presented. The new formulation is free of any approximation. The obtained semidefinite program is reduced to a linear program where its numerical solution is feasible for reasonable message lengths. Simulations confirm that the optimized degree distributions outperform Robust Soliton distribution, both in terms of overhead and Encoding/Decoding complexity.