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XOR network-based on-chip test compression schemes have been widely employed in large industrial scan designs due to their high compression ratio and efficient decompression mechanism. Nevertheless, such a scheme necessitates high unspecified bit ratios in the original test cubes, resulting in quite significant difficulties in preprocessing test cubes for scan power reduction. The linear mapping from the original cubes to the compressed seeds typically provides extra degrees of flexibility as multiple seeds may reconstruct the test cube. Due to the highly divergent power impact of distinct seeds though, appreciable power reductions in the decompressed test data can be attained through the pinpointing of the power-optimal seeds during the compression phase. This work explores the aforementioned flexibility in the seed space, and outlines a mathematical and algorithmic framework for a power-aware linear test compression scheme. The proposed technique incurs no hardware overhead over the traditional linear compression scheme; it can be easily embedded furthermore into the industrial test compaction/compression flow. Experimental results confirm that the proposed technique delivers significant scan power reduction with negligible impact on the compression ratio.