We consider a binary energy harvesting communication system with a finite battery transmitter over a noisy channel, and design explicit and implementable codes based on concatenation of a nonlinear trellis code (NLTC) with an outer low density parity check (LDPC) code. We propose two different decoding methods where the simplified one ignores the memory in the battery state while the more sophisticated one utilizes the memory. Numerical results demonstrate that the designed codes outperform other reference schemes. The results also show the superiority of the improved decoding approach over the naive solution.