This paper investigates the puncturing pattern design of Hadamard codes. When designing the puncturing patterns, the minimum Hamming distance property and cross-correlation property of the punctured Hadamard codes are considered. The theoretical upper bound of the minimum Hamming distance is derived for the punctured Hadamard codes and an algorithm is subsequently proposed for searching punctured codes with the minimum distance approaching the bound. Another approach to design the punctured codes is to minimize the cross-correlations between punctured codewords. A new metric is defined to measure the overall severity of the cross-correlation based on which punctured codes with the 'lowest' cross-correlations are found. Simulation results show that both techniques are effective in searching for good punctured Hadamard codes. As expected, moreover, the punctured codes found by minimizing the cross-correlation outperform those optimized by maximizing minimum Hamming distance when a-posteriori-probability decoding is used.