In this paper, we consider the construction of a Slepian-Wolf source coding scheme in a context where only a small amount of data has to be transmitted to the decoder. In this context, we propose a novel rate-adaptive Slepian-Wolf code construction that is based on non-binary LDPC codes. The construction we propose replaces the regular accumulator of the standard LDPCA method by a local graph that is optimized at every rate of interest. In our method, the local graph is specifically designed in order to give good decoding performance at short length, while existing LDPCA constructions are usually optimized under an infinite codeword length assumption. Our simulation results on short codes obtained from our design method show a FER improvement of up to an order to magnitude compared to the standard LDPCA construction.