Calculating the minimum distance of linear block codes via Integer Programming

In recent years several authors have used Integer Programming (IP) formulations to model the Maximum Likelihood (ML) decoding problem. IP formulation of the ML decoding problem can be modified to calculate the minimum distance of the code also. The IP based approach for MD calculation can be effectively used for small block length codes. However, it is not known if this approach is useful for the moderate and long length codes used in practice. In this paper we demonstrate the usefulness of the IP based approach for the MD calculation by applying it to practically relevant codes. We calculate either the true MD or upper bounds on the MD for 3GPP LTE turbo codes, WiMax duo-binary turbo codes and WiMax & WiFi low density parity-check codes. A new IP formulation is also introduced to calculate the MD of punctured turbo codes. Along with MD calculation, the same IP formulation is used to find multiplicity of low weight codewords. Our results prove that, the IP based approach can estimate the MD of practically relevant codes fairly quickly and can also provide information about the multiplicity of codewords with a given Hamming weight. Also, the IP based method for the MD calculation is independent of the code used and can easily accommodate puncturing schemes.