Let C be a perfectly smooth LDC that encodes n-bit messages to N-bit codewords. As a preprocessing step, each of the k servers S_1,\ldots ,S_k encodes the n-bit database x with the code C, so each server now stores the N-bit codeword C(x). A user interested in obtaining the i^{th} bit of x randomly generates a set of k queries q_1,\ldots q_k such that x_i can be computed from C(x)_{q_1}, \ldots C(x)_{q_k} using the local decoding algorithm A for C. The user sends each query to a different server, and each server responds with the bit requested. The user then uses A to compute x_i from the responses.
