Topological subsystem codes were proposed by Bombin based on 3-face-colorablecubic graphs. Suchara, Bravyi and Terhal generalized this construction andproposed a method to construct topological subsystem codes using 3-valenthypergraphs that satisfy certain constraints. Finding such hypergraphs andcomputing their parameters however is a nontrivial task. We propose families oftopological subsystem codes that were previously not known. In particular, ourconstructions give codes which cannot be derived from Bombin's construction. Wealso study the error recovery schemes for the proposed subsystem codes and givedetailed schedules for the syndrome measurement that take advantage of the2-locality of the gauge group. The study also leads to a new and generalconstruction for color codes.
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