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For a given polar code construction, the existing literature on puncturing for polar codes focuses on finding the optimal puncturing pattern, and then re-selecting the information set. This paper devotes itself to find the optimal puncturing pattern when the information set is fixed. Puncturing the coded bits corresponding to the worst quality bit channels, called worst quality puncturing (WQP), is proposed, which is analyzed to minimize the bit channel quality loss at the punctured positions. The simulation results show that the WQP outperforms the best existing puncturing schemes when the information set is fixed.