In this paper, we use a linear programming (LP) optimization approach to evaluate the equivocation when coding over a wiretap channel model where the main channel is noiseless and the eavesdropper's channel is a binary symmetric channel (BSC). Using this technique, we present a numerically-derived upper bound for the achievable secrecy rate in the finite blocklength regime that is tighter than traditional infinite blocklength bounds. We also propose a secrecy coding technique that outperforms random binning codes. When there is one overhead bit, this coding technique is optimum and achieves the newly derived bound. For cases with additional bits of overhead, our coding scheme can achieve equivocation rates close to the new bound. Furthermore, we explore the patterns of the generator matrix and the parity-check matrix for linear codes and we present binning techniques for both linear and nonlinear codes using two different approaches: recursive and non-recursive. To our knowledge, this is the first optimization solution for secrecy coding obtained through linear programming. Our new bounds and codes mark a significant breakthrough towards understanding fundamental limits of performance (and how to achieve them in some instances) for the binary symmetric wiretap channel with real finite blocklength coding constructions. Our techniques are especially useful for codes of small to medium blocklength, such as those that may be required by applications with small payloads, such as the Internet of Things.