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Ternary content addressable memories (TCAMs) have become the de facto standard in industry for fast packet classification. Unfortunately, TCAMs have limitations of small capacity, high power consumption, high heat generation, and high cost. The well-known range expansion problem exacerbates these limitations as each classifier rule typically has to be converted to multiple TCAM rules. One method for coping with these limitations is to use compression schemes to reduce the number of TCAM rules required to represent a classifier. Unfortunately, all existing compression schemes only produce prefix classifiers. Thus, they all miss the compression opportunities created by non-prefix ternary classifiers. In this paper, we propose bit weaving, the first non-prefix compression scheme. Bit weaving is based on the observation that TCAM entries that have the same decision and whose predicates differ by only one bit can be merged into one entry by replacing the bit in question with . Bit weaving consists of two new techniques, bit swapping and bit merging, to first identify and then merge such rules together. The key advantages of bit weaving are that it runs fast, it is effective, and it is composable with other TCAM optimization methods as a pre/post-processing routine. We implemented bit weaving and conducted experiments on both real-world and synthetic packet classifiers. Our experimental results show the following: 1) bit weaving is an effective standalone compression technique (it achieves an average compression ratio of 23.6%); 2) bit weaving finds compression opportunities that other methods miss. Specifically, bit weaving improves the prior TCAM optimization techniques of TCAM Razor and Topological Transformation by an average of 12.8% and 36.5%, respectively.