{"id":"1276.3.2.a","row":1368,"stored":{"file":"master_catalog.json","row":1368,"construction":null,"columns_sha256":null},"parameters":{"n":1276,"k":3,"d":2,"N":12,"r":9,"level":3},"gate":{"string":"01+02+12+012","human":"CS01\u00b7CS02\u00b7CS12\u00b7CCZ012","monomials":[[0,1],[0,2],[1,2],[0,1,2]],"sk_key":[[0,1],[0,1,2],[0,2],[1,2]],"sk_fingerprint":"68b62311d67769e3","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,1,2,3,4],[0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9,10],[0,1,2,3,4,5,6,7,8,9,11],[0,1,2,3,4,5,6,7,8,10,11],[0,1,2,3,4,5,6,7,9],[0,1,2,3,4,5,6,7,9,10,11],[0,1,2,3,4,5,6,7,10],[0,1,2,3,4,5,6,7,11],[0,1,2,3,4,5,6,8,9],[0,1,2,3,4,5,6,8,9,10,11],[0,1,2,3,4,5,6,8,10],[0,1,2,3,4,5,6,8,11],[0,1,2,3,4,5,6,9,10],[0,1,2,3,4,5,6,9,11],[0,1,2,3,4,5,6,10,11],[0,1,2,3,4,5,7],[0,1,2,3,4,5,7,8,9],[0,1,2,3,4,5,7,8,9,10,11],[0,1,2,3,4,5,7,8,10],[0,1,2,3,4,5,7,8,11],[0,1,2,3,4,5,7,9,10],[0,1,2,3,4,5,7,9,11],[0,1,2,3,4,5,7,10,11],[0,1,2,3,4,5,8],[0,1,2,3,4,5,8,9,10],[0,1,2,3,4,5,8,9,11],[0,1,2,3,4,5,8,10,11],[0,1,2,3,4,5,9],[0,1,2,3,4,5,9,10,11],[0,1,2,3,4,5,10],[0,1,2,3,4,5,11],[0,1,2,3,4,6,7],[0,1,2,3,4,6,7,8,9],[0,1,2,3,4,6,7,8,9,10,11],[0,1,2,3,4,6,7,8,10],[0,1,2,3,4,6,7,8,11],[0,1,2,3,4,6,7,9,10],[0,1,2,3,4,6,7,9,11],[0,1,2,3,4,6,7,10,11],[0,1,2,3,4,6,8],[0,1,2,3,4,6,8,9,10],[0,1,2,3,4,6,8,9,11],[0,1,2,3,4,6,8,10,11],[0,1,2,3,4,6,9],[0,1,2,3,4,6,9,10,11],[0,1,2,3,4,6,10],[0,1,2,3,4,6,11],[0,1,2,3,4,7,8],[0,1,2,3,4,7,8,9,10],[0,1,2,3,4,7,8,9,11],[0,1,2,3,4,7,8,10,11],[0,1,2,3,4,7,9],[0,1,2,3,4,7,9,10,11],[0,1,2,3,4,7,10],[0,1,2,3,4,7,11],[0,1,2,3,4,8,9],[0,1,2,3,4,8,9,10,11],[0,1,2,3,4,8,10],[0,1,2,3,4,8,11],[0,1,2,3,4,9,10],[0,1,2,3,4,9,11],[0,1,2,3,4,10,11],[0,1,2,3,5],[0,1,2,3,5,6,7],[0,1,2,3,5,6,7,8,9],[0,1,2,3,5,6,7,8,9,10,11],[0,1,2,3,5,6,7,8,10],[0,1,2,3,5,6,7,8,11],[0,1,2,3,5,6,7,9,10],[0,1,2,3,5,6,7,9,11],[0,1,2,3,5,6,7,10,11],[0,1,2,3,5,6,8],[0,1,2,3,5,6,8,9,10],[0,1,2,3,5,6,8,9,11],[0,1,2,3,5,6,8,10,11],[0,1,2,3,5,6,9],[0,1,2,3,5,6,9,10,11],[0,1,2,3,5,6,10],[0,1,2,3,5,6,11],[0,1,2,3,5,7,8],[0,1,2,3,5,7,8,9,10],[0,1,2,3,5,7,8,9,11],[0,1,2,3,5,7,8,10,11],[0,1,2,3,5,7,9],[0,1,2,3,5,7,9,10,11],[0,1,2,3,5,7,10],[0,1,2,3,5,7,11],[0,1,2,3,5,8,9],[0,1,2,3,5,8,9,10,11],[0,1,2,3,5,8,10],[0,1,2,3,5,8,11],[0,1,2,3,5,9,10],[0,1,2,3,5,9,11],[0,1,2,3,5,10,11],[0,1,2,3,6],[0,1,2,3,6,7,8],[0,1,2,3,6,7,8,9,10],[0,1,2,3,6,7,8,9,11],[0,1,2,3,6,7,8,10,11],[0,1,2,3,6,7,9],[0,1,2,3,6,7,9,10,11],[0,1,2,3,6,7,10],[0,1,2,3,6,7,11],[0,1,2,3,6,8,9],[0,1,2,3,6,8,9,10,11],[0,1,2,3,6,8,10],[0,1,2,3,6,8,11],[0,1,2,3,6,9,10],[0,1,2,3,6,9,11],[0,1,2,3,6,10,11],[0,1,2,3,7],[0,1,2,3,7,8,9],[0,1,2,3,7,8,9,10,11],[0,1,2,3,7,8,10],[0,1,2,3,7,8,11],[0,1,2,3,7,9,10],[0,1,2,3,7,9,11],[0,1,2,3,7,10,11],[0,1,2,3,8],[0,1,2,3,8,9,10],[0,1,2,3,8,9,11],[0,1,2,3,8,10,11],[0,1,2,3,9],[0,1,2,3,9,10,11],[0,1,2,3,10],[0,1,2,3,11],[0,1,2,4,5],[0,1,2,4,5,6,7],[0,1,2,4,5,6,7,8,9],[0,1,2,4,5,6,7,8,9,10,11],[0,1,2,4,5,6,7,8,10],[0,1,2,4,5,6,7,8,11],[0,1,2,4,5,6,7,9,10],[0,1,2,4,5,6,7,9,11],[0,1,2,4,5,6,7,10,11],[0,1,2,4,5,6,8],[0,1,2,4,5,6,8,9,10],[0,1,2,4,5,6,8,9,11],[0,1,2,4,5,6,8,10,11],[0,1,2,4,5,6,9],[0,1,2,4,5,6,9,10,11],[0,1,2,4,5,6,10],[0,1,2,4,5,6,11],[0,1,2,4,5,7,8],[0,1,2,4,5,7,8,9,10],[0,1,2,4,5,7,8,9,11],[0,1,2,4,5,7,8,10,11],[0,1,2,4,5,7,9],[0,1,2,4,5,7,9,10,11],[0,1,2,4,5,7,10],[0,1,2,4,5,7,11],[0,1,2,4,5,8,9],[0,1,2,4,5,8,9,10,11],[0,1,2,4,5,8,10],[0,1,2,4,5,8,11],[0,1,2,4,5,9,10],[0,1,2,4,5,9,11],[0,1,2,4,5,10,11],[0,1,2,4,6],[0,1,2,4,6,7,8],[0,1,2,4,6,7,8,9,10],[0,1,2,4,6,7,8,9,11],[0,1,2,4,6,7,8,10,11],[0,1,2,4,6,7,9],[0,1,2,4,6,7,9,10,11],[0,1,2,4,6,7,10],[0,1,2,4,6,7,11],[0,1,2,4,6,8,9],[0,1,2,4,6,8,9,10,11],[0,1,2,4,6,8,10],[0,1,2,4,6,8,11],[0,1,2,4,6,9,10],[0,1,2,4,6,9,11],[0,1,2,4,6,10,11],[0,1,2,4,7],[0,1,2,4,7,8,9],[0,1,2,4,7,8,9,10,11],[0,1,2,4,7,8,10],[0,1,2,4,7,8,11],[0,1,2,4,7,9,10],[0,1,2,4,7,9,11],[0,1,2,4,7,10,11],[0,1,2,4,8],[0,1,2,4,8,9,10],[0,1,2,4,8,9,11],[0,1,2,4,8,10,11],[0,1,2,4,9],[0,1,2,4,9,10,11],[0,1,2,4,10],[0,1,2,4,11],[0,1,2,5,6],[0,1,2,5,6,7,8],[0,1,2,5,6,7,8,9,10],[0,1,2,5,6,7,8,9,11],[0,1,2,5,6,7,8,10,11],[0,1,2,5,6,7,9],[0,1,2,5,6,7,9,10,11],[0,1,2,5,6,7,10],[0,1,2,5,6,7,11],[0,1,2,5,6,8,9],[0,1,2,5,6,8,9,10,11],[0,1,2,5,6,8,10],[0,1,2,5,6,8,11],[0,1,2,5,6,9,10],[0,1,2,5,6,9,11],[0,1,2,5,6,10,11],[0,1,2,5,7],[0,1,2,5,7,8,9],[0,1,2,5,7,8,9,10,11],[0,1,2,5,7,8,10],[0,1,2,5,7,8,11],[0,1,2,5,7,9,10],[0,1,2,5,7,9,11],[0,1,2,5,7,10,11],[0,1,2,5,8],[0,1,2,5,8,9,10],[0,1,2,5,8,9,11],[0,1,2,5,8,10,11],[0,1,2,5,9],[0,1,2,5,9,10,11],[0,1,2,5,10],[0,1,2,5,11],[0,1,2,6,7],[0,1,2,6,7,8,9],[0,1,2,6,7,8,9,10,11],[0,1,2,6,7,8,10],[0,1,2,6,7,8,11],[0,1,2,6,7,9,10],[0,1,2,6,7,9,11],[0,1,2,6,7,10,11],[0,1,2,6,8],[0,1,2,6,8,9,10],[0,1,2,6,8,9,11],[0,1,2,6,8,10,11],[0,1,2,6,9],[0,1,2,6,9,10,11],[0,1,2,6,10],[0,1,2,6,11],[0,1,2,7,8],[0,1,2,7,8,9,10],[0,1,2,7,8,9,11],[0,1,2,7,8,10,11],[0,1,2,7,9],[0,1,2,7,9,10,11],[0,1,2,7,10],[0,1,2,7,11],[0,1,2,8,9],[0,1,2,8,9,10,11],[0,1,2,8,10],[0,1,2,8,11],[0,1,2,9,10],[0,1,2,9,11],[0,1,2,10,11],[0,3,4],[0,3,4,5,6],[0,3,4,5,6,7,8],[0,3,4,5,6,7,8,9,10],[0,3,4,5,6,7,8,9,11],[0,3,4,5,6,7,8,10,11],[0,3,4,5,6,7,9],[0,3,4,5,6,7,9,10,11],[0,3,4,5,6,7,10],[0,3,4,5,6,7,11],[0,3,4,5,6,8,9],[0,3,4,5,6,8,9,10,11],[0,3,4,5,6,8,10],[0,3,4,5,6,8,11],[0,3,4,5,6,9,10],[0,3,4,5,6,9,11],[0,3,4,5,6,10,11],[0,3,4,5,7],[0,3,4,5,7,8,9],[0,3,4,5,7,8,9,10,11],[0,3,4,5,7,8,10],[0,3,4,5,7,8,11],[0,3,4,5,7,9,10],[0,3,4,5,7,9,11],[0,3,4,5,7,10,11],[0,3,4,5,8],[0,3,4,5,8,9,10],[0,3,4,5,8,9,11],[0,3,4,5,8,10,11],[0,3,4,5,9],[0,3,4,5,9,10,11],[0,3,4,5,10],[0,3,4,5,11],[0,3,4,6,7],[0,3,4,6,7,8,9],[0,3,4,6,7,8,9,10,11],[0,3,4,6,7,8,10],[0,3,4,6,7,8,11],[0,3,4,6,7,9,10],[0,3,4,6,7,9,11],[0,3,4,6,7,10,11],[0,3,4,6,8],[0,3,4,6,8,9,10],[0,3,4,6,8,9,11],[0,3,4,6,8,10,11],[0,3,4,6,9],[0,3,4,6,9,10,11],[0,3,4,6,10],[0,3,4,6,11],[0,3,4,7,8],[0,3,4,7,8,9,10],[0,3,4,7,8,9,11],[0,3,4,7,8,10,11],[0,3,4,7,9],[0,3,4,7,9,10,11],[0,3,4,7,10],[0,3,4,7,11],[0,3,4,8,9],[0,3,4,8,9,10,11],[0,3,4,8,10],[0,3,4,8,11],[0,3,4,9,10],[0,3,4,9,11],[0,3,4,10,11],[0,3,5],[0,3,5,6,7],[0,3,5,6,7,8,9],[0,3,5,6,7,8,9,10,11],[0,3,5,6,7,8,10],[0,3,5,6,7,8,11],[0,3,5,6,7,9,10],[0,3,5,6,7,9,11],[0,3,5,6,7,10,11],[0,3,5,6,8],[0,3,5,6,8,9,10],[0,3,5,6,8,9,11],[0,3,5,6,8,10,11],[0,3,5,6,9],[0,3,5,6,9,10,11],[0,3,5,6,10],[0,3,5,6,11],[0,3,5,7,8],[0,3,5,7,8,9,10],[0,3,5,7,8,9,11],[0,3,5,7,8,10,11],[0,3,5,7,9],[0,3,5,7,9,10,11],[0,3,5,7,10],[0,3,5,7,11],[0,3,5,8,9],[0,3,5,8,9,10,11],[0,3,5,8,10],[0,3,5,8,11],[0,3,5,9,10],[0,3,5,9,11],[0,3,5,10,11],[0,3,6],[0,3,6,7,8],[0,3,6,7,8,9,10],[0,3,6,7,8,9,11],[0,3,6,7,8,10,11],[0,3,6,7,9],[0,3,6,7,9,10,11],[0,3,6,7,10],[0,3,6,7,11],[0,3,6,8,9],[0,3,6,8,9,10,11],[0,3,6,8,10],[0,3,6,8,11],[0,3,6,9,10],[0,3,6,9,11],[0,3,6,10,11],[0,3,7],[0,3,7,8,9],[0,3,7,8,9,10,11],[0,3,7,8,10],[0,3,7,8,11],[0,3,7,9,10],[0,3,7,9,11],[0,3,7,10,11],[0,3,8],[0,3,8,9,10],[0,3,8,9,11],[0,3,8,10,11],[0,3,9],[0,3,9,10,11],[0,3,10],[0,3,11],[0,4,5],[0,4,5,6,7],[0,4,5,6,7,8,9],[0,4,5,6,7,8,9,10,11],[0,4,5,6,7,8,10],[0,4,5,6,7,8,11],[0,4,5,6,7,9,10],[0,4,5,6,7,9,11],[0,4,5,6,7,10,11],[0,4,5,6,8],[0,4,5,6,8,9,10],[0,4,5,6,8,9,11],[0,4,5,6,8,10,11],[0,4,5,6,9],[0,4,5,6,9,10,11],[0,4,5,6,10],[0,4,5,6,11],[0,4,5,7,8],[0,4,5,7,8,9,10],[0,4,5,7,8,9,11],[0,4,5,7,8,10,11],[0,4,5,7,9],[0,4,5,7,9,10,11],[0,4,5,7,10],[0,4,5,7,11],[0,4,5,8,9],[0,4,5,8,9,10,11],[0,4,5,8,10],[0,4,5,8,11],[0,4,5,9,10],[0,4,5,9,11],[0,4,5,10,11],[0,4,6],[0,4,6,7,8],[0,4,6,7,8,9,10],[0,4,6,7,8,9,11],[0,4,6,7,8,10,11],[0,4,6,7,9],[0,4,6,7,9,10,11],[0,4,6,7,10],[0,4,6,7,11],[0,4,6,8,9],[0,4,6,8,9,10,11],[0,4,6,8,10],[0,4,6,8,11],[0,4,6,9,10],[0,4,6,9,11],[0,4,6,10,11],[0,4,7],[0,4,7,8,9],[0,4,7,8,9,10,11],[0,4,7,8,10],[0,4,7,8,11],[0,4,7,9,10],[0,4,7,9,11],[0,4,7,10,11],[0,4,8],[0,4,8,9,10],[0,4,8,9,11],[0,4,8,10,11],[0,4,9],[0,4,9,10,11],[0,4,10],[0,4,11],[0,5,6],[0,5,6,7,8],[0,5,6,7,8,9,10],[0,5,6,7,8,9,11],[0,5,6,7,8,10,11],[0,5,6,7,9],[0,5,6,7,9,10,11],[0,5,6,7,10],[0,5,6,7,11],[0,5,6,8,9],[0,5,6,8,9,10,11],[0,5,6,8,10],[0,5,6,8,11],[0,5,6,9,10],[0,5,6,9,11],[0,5,6,10,11],[0,5,7],[0,5,7,8,9],[0,5,7,8,9,10,11],[0,5,7,8,10],[0,5,7,8,11],[0,5,7,9,10],[0,5,7,9,11],[0,5,7,10,11],[0,5,8],[0,5,8,9,10],[0,5,8,9,11],[0,5,8,10,11],[0,5,9],[0,5,9,10,11],[0,5,10],[0,5,11],[0,6,7],[0,6,7,8,9],[0,6,7,8,9,10,11],[0,6,7,8,10],[0,6,7,8,11],[0,6,7,9,10],[0,6,7,9,11],[0,6,7,10,11],[0,6,8],[0,6,8,9,10],[0,6,8,9,11],[0,6,8,10,11],[0,6,9],[0,6,9,10,11],[0,6,10],[0,6,11],[0,7,8],[0,7,8,9,10],[0,7,8,9,11],[0,7,8,10,11],[0,7,9],[0,7,9,10,11],[0,7,10],[0,7,11],[0,8,9],[0,8,9,10,11],[0,8,10],[0,8,11],[0,9,10],[0,9,11],[0,10,11],[1,3,4],[1,3,4,5,6],[1,3,4,5,6,7,8],[1,3,4,5,6,7,8,9,10],[1,3,4,5,6,7,8,9,11],[1,3,4,5,6,7,8,10,11],[1,3,4,5,6,7,9],[1,3,4,5,6,7,9,10,11],[1,3,4,5,6,7,10],[1,3,4,5,6,7,11],[1,3,4,5,6,8,9],[1,3,4,5,6,8,9,10,11],[1,3,4,5,6,8,10],[1,3,4,5,6,8,11],[1,3,4,5,6,9,10],[1,3,4,5,6,9,11],[1,3,4,5,6,10,11],[1,3,4,5,7],[1,3,4,5,7,8,9],[1,3,4,5,7,8,9,10,11],[1,3,4,5,7,8,10],[1,3,4,5,7,8,11],[1,3,4,5,7,9,10],[1,3,4,5,7,9,11],[1,3,4,5,7,10,11],[1,3,4,5,8],[1,3,4,5,8,9,10],[1,3,4,5,8,9,11],[1,3,4,5,8,10,11],[1,3,4,5,9],[1,3,4,5,9,10,11],[1,3,4,5,10],[1,3,4,5,11],[1,3,4,6,7],[1,3,4,6,7,8,9],[1,3,4,6,7,8,9,10,11],[1,3,4,6,7,8,10],[1,3,4,6,7,8,11],[1,3,4,6,7,9,10],[1,3,4,6,7,9,11],[1,3,4,6,7,10,11],[1,3,4,6,8],[1,3,4,6,8,9,10],[1,3,4,6,8,9,11],[1,3,4,6,8,10,11],[1,3,4,6,9],[1,3,4,6,9,10,11],[1,3,4,6,10],[1,3,4,6,11],[1,3,4,7,8],[1,3,4,7,8,9,10],[1,3,4,7,8,9,11],[1,3,4,7,8,10,11],[1,3,4,7,9],[1,3,4,7,9,10,11],[1,3,4,7,10],[1,3,4,7,11],[1,3,4,8,9],[1,3,4,8,9,10,11],[1,3,4,8,10],[1,3,4,8,11],[1,3,4,9,10],[1,3,4,9,11],[1,3,4,10,11],[1,3,5],[1,3,5,6,7],[1,3,5,6,7,8,9],[1,3,5,6,7,8,9,10,11],[1,3,5,6,7,8,10],[1,3,5,6,7,8,11],[1,3,5,6,7,9,10],[1,3,5,6,7,9,11],[1,3,5,6,7,10,11],[1,3,5,6,8],[1,3,5,6,8,9,10],[1,3,5,6,8,9,11],[1,3,5,6,8,10,11],[1,3,5,6,9],[1,3,5,6,9,10,11],[1,3,5,6,10],[1,3,5,6,11],[1,3,5,7,8],[1,3,5,7,8,9,10],[1,3,5,7,8,9,11],[1,3,5,7,8,10,11],[1,3,5,7,9],[1,3,5,7,9,10,11],[1,3,5,7,10],[1,3,5,7,11],[1,3,5,8,9],[1,3,5,8,9,10,11],[1,3,5,8,10],[1,3,5,8,11],[1,3,5,9,10],[1,3,5,9,11],[1,3,5,10,11],[1,3,6],[1,3,6,7,8],[1,3,6,7,8,9,10],[1,3,6,7,8,9,11],[1,3,6,7,8,10,11],[1,3,6,7,9],[1,3,6,7,9,10,11],[1,3,6,7,10],[1,3,6,7,11],[1,3,6,8,9],[1,3,6,8,9,10,11],[1,3,6,8,10],[1,3,6,8,11],[1,3,6,9,10],[1,3,6,9,11],[1,3,6,10,11],[1,3,7],[1,3,7,8,9],[1,3,7,8,9,10,11],[1,3,7,8,10],[1,3,7,8,11],[1,3,7,9,10],[1,3,7,9,11],[1,3,7,10,11],[1,3,8],[1,3,8,9,10],[1,3,8,9,11],[1,3,8,10,11],[1,3,9],[1,3,9,10,11],[1,3,10],[1,3,11],[1,4,5],[1,4,5,6,7],[1,4,5,6,7,8,9],[1,4,5,6,7,8,9,10,11],[1,4,5,6,7,8,10],[1,4,5,6,7,8,11],[1,4,5,6,7,9,10],[1,4,5,6,7,9,11],[1,4,5,6,7,10,11],[1,4,5,6,8],[1,4,5,6,8,9,10],[1,4,5,6,8,9,11],[1,4,5,6,8,10,11],[1,4,5,6,9],[1,4,5,6,9,10,11],[1,4,5,6,10],[1,4,5,6,11],[1,4,5,7,8],[1,4,5,7,8,9,10],[1,4,5,7,8,9,11],[1,4,5,7,8,10,11],[1,4,5,7,9],[1,4,5,7,9,10,11],[1,4,5,7,10],[1,4,5,7,11],[1,4,5,8,9],[1,4,5,8,9,10,11],[1,4,5,8,10],[1,4,5,8,11],[1,4,5,9,10],[1,4,5,9,11],[1,4,5,10,11],[1,4,6],[1,4,6,7,8],[1,4,6,7,8,9,10],[1,4,6,7,8,9,11],[1,4,6,7,8,10,11],[1,4,6,7,9],[1,4,6,7,9,10,11],[1,4,6,7,10],[1,4,6,7,11],[1,4,6,8,9],[1,4,6,8,9,10,11],[1,4,6,8,10],[1,4,6,8,11],[1,4,6,9,10],[1,4,6,9,11],[1,4,6,10,11],[1,4,7],[1,4,7,8,9],[1,4,7,8,9,10,11],[1,4,7,8,10],[1,4,7,8,11],[1,4,7,9,10],[1,4,7,9,11],[1,4,7,10,11],[1,4,8],[1,4,8,9,10],[1,4,8,9,11],[1,4,8,10,11],[1,4,9],[1,4,9,10,11],[1,4,10],[1,4,11],[1,5,6],[1,5,6,7,8],[1,5,6,7,8,9,10],[1,5,6,7,8,9,11],[1,5,6,7,8,10,11],[1,5,6,7,9],[1,5,6,7,9,10,11],[1,5,6,7,10],[1,5,6,7,11],[1,5,6,8,9],[1,5,6,8,9,10,11],[1,5,6,8,10],[1,5,6,8,11],[1,5,6,9,10],[1,5,6,9,11],[1,5,6,10,11],[1,5,7],[1,5,7,8,9],[1,5,7,8,9,10,11],[1,5,7,8,10],[1,5,7,8,11],[1,5,7,9,10],[1,5,7,9,11],[1,5,7,10,11],[1,5,8],[1,5,8,9,10],[1,5,8,9,11],[1,5,8,10,11],[1,5,9],[1,5,9,10,11],[1,5,10],[1,5,11],[1,6,7],[1,6,7,8,9],[1,6,7,8,9,10,11],[1,6,7,8,10],[1,6,7,8,11],[1,6,7,9,10],[1,6,7,9,11],[1,6,7,10,11],[1,6,8],[1,6,8,9,10],[1,6,8,9,11],[1,6,8,10,11],[1,6,9],[1,6,9,10,11],[1,6,10],[1,6,11],[1,7,8],[1,7,8,9,10],[1,7,8,9,11],[1,7,8,10,11],[1,7,9],[1,7,9,10,11],[1,7,10],[1,7,11],[1,8,9],[1,8,9,10,11],[1,8,10],[1,8,11],[1,9,10],[1,9,11],[1,10,11],[2,3,4],[2,3,4,5,6],[2,3,4,5,6,7,8],[2,3,4,5,6,7,8,9,10],[2,3,4,5,6,7,8,9,11],[2,3,4,5,6,7,8,10,11],[2,3,4,5,6,7,9],[2,3,4,5,6,7,9,10,11],[2,3,4,5,6,7,10],[2,3,4,5,6,7,11],[2,3,4,5,6,8,9],[2,3,4,5,6,8,9,10,11],[2,3,4,5,6,8,10],[2,3,4,5,6,8,11],[2,3,4,5,6,9,10],[2,3,4,5,6,9,11],[2,3,4,5,6,10,11],[2,3,4,5,7],[2,3,4,5,7,8,9],[2,3,4,5,7,8,9,10,11],[2,3,4,5,7,8,10],[2,3,4,5,7,8,11],[2,3,4,5,7,9,10],[2,3,4,5,7,9,11],[2,3,4,5,7,10,11],[2,3,4,5,8],[2,3,4,5,8,9,10],[2,3,4,5,8,9,11],[2,3,4,5,8,10,11],[2,3,4,5,9],[2,3,4,5,9,10,11],[2,3,4,5,10],[2,3,4,5,11],[2,3,4,6,7],[2,3,4,6,7,8,9],[2,3,4,6,7,8,9,10,11],[2,3,4,6,7,8,10],[2,3,4,6,7,8,11],[2,3,4,6,7,9,10],[2,3,4,6,7,9,11],[2,3,4,6,7,10,11],[2,3,4,6,8],[2,3,4,6,8,9,10],[2,3,4,6,8,9,11],[2,3,4,6,8,10,11],[2,3,4,6,9],[2,3,4,6,9,10,11],[2,3,4,6,10],[2,3,4,6,11],[2,3,4,7,8],[2,3,4,7,8,9,10],[2,3,4,7,8,9,11],[2,3,4,7,8,10,11],[2,3,4,7,9],[2,3,4,7,9,10,11],[2,3,4,7,10],[2,3,4,7,11],[2,3,4,8,9],[2,3,4,8,9,10,11],[2,3,4,8,10],[2,3,4,8,11],[2,3,4,9,10],[2,3,4,9,11],[2,3,4,10,11],[2,3,5],[2,3,5,6,7],[2,3,5,6,7,8,9],[2,3,5,6,7,8,9,10,11],[2,3,5,6,7,8,10],[2,3,5,6,7,8,11],[2,3,5,6,7,9,10],[2,3,5,6,7,9,11],[2,3,5,6,7,10,11],[2,3,5,6,8],[2,3,5,6,8,9,10],[2,3,5,6,8,9,11],[2,3,5,6,8,10,11],[2,3,5,6,9],[2,3,5,6,9,10,11],[2,3,5,6,10],[2,3,5,6,11],[2,3,5,7,8],[2,3,5,7,8,9,10],[2,3,5,7,8,9,11],[2,3,5,7,8,10,11],[2,3,5,7,9],[2,3,5,7,9,10,11],[2,3,5,7,10],[2,3,5,7,11],[2,3,5,8,9],[2,3,5,8,9,10,11],[2,3,5,8,10],[2,3,5,8,11],[2,3,5,9,10],[2,3,5,9,11],[2,3,5,10,11],[2,3,6],[2,3,6,7,8],[2,3,6,7,8,9,10],[2,3,6,7,8,9,11],[2,3,6,7,8,10,11],[2,3,6,7,9],[2,3,6,7,9,10,11],[2,3,6,7,10],[2,3,6,7,11],[2,3,6,8,9],[2,3,6,8,9,10,11],[2,3,6,8,10],[2,3,6,8,11],[2,3,6,9,10],[2,3,6,9,11],[2,3,6,10,11],[2,3,7],[2,3,7,8,9],[2,3,7,8,9,10,11],[2,3,7,8,10],[2,3,7,8,11],[2,3,7,9,10],[2,3,7,9,11],[2,3,7,10,11],[2,3,8],[2,3,8,9,10],[2,3,8,9,11],[2,3,8,10,11],[2,3,9],[2,3,9,10,11],[2,3,10],[2,3,11],[2,4,5],[2,4,5,6,7],[2,4,5,6,7,8,9],[2,4,5,6,7,8,9,10,11],[2,4,5,6,7,8,10],[2,4,5,6,7,8,11],[2,4,5,6,7,9,10],[2,4,5,6,7,9,11],[2,4,5,6,7,10,11],[2,4,5,6,8],[2,4,5,6,8,9,10],[2,4,5,6,8,9,11],[2,4,5,6,8,10,11],[2,4,5,6,9],[2,4,5,6,9,10,11],[2,4,5,6,10],[2,4,5,6,11],[2,4,5,7,8],[2,4,5,7,8,9,10],[2,4,5,7,8,9,11],[2,4,5,7,8,10,11],[2,4,5,7,9],[2,4,5,7,9,10,11],[2,4,5,7,10],[2,4,5,7,11],[2,4,5,8,9],[2,4,5,8,9,10,11],[2,4,5,8,10],[2,4,5,8,11],[2,4,5,9,10],[2,4,5,9,11],[2,4,5,10,11],[2,4,6],[2,4,6,7,8],[2,4,6,7,8,9,10],[2,4,6,7,8,9,11],[2,4,6,7,8,10,11],[2,4,6,7,9],[2,4,6,7,9,10,11],[2,4,6,7,10],[2,4,6,7,11],[2,4,6,8,9],[2,4,6,8,9,10,11],[2,4,6,8,10],[2,4,6,8,11],[2,4,6,9,10],[2,4,6,9,11],[2,4,6,10,11],[2,4,7],[2,4,7,8,9],[2,4,7,8,9,10,11],[2,4,7,8,10],[2,4,7,8,11],[2,4,7,9,10],[2,4,7,9,11],[2,4,7,10,11],[2,4,8],[2,4,8,9,10],[2,4,8,9,11],[2,4,8,10,11],[2,4,9],[2,4,9,10,11],[2,4,10],[2,4,11],[2,5,6],[2,5,6,7,8],[2,5,6,7,8,9,10],[2,5,6,7,8,9,11],[2,5,6,7,8,10,11],[2,5,6,7,9],[2,5,6,7,9,10,11],[2,5,6,7,10],[2,5,6,7,11],[2,5,6,8,9],[2,5,6,8,9,10,11],[2,5,6,8,10],[2,5,6,8,11],[2,5,6,9,10],[2,5,6,9,11],[2,5,6,10,11],[2,5,7],[2,5,7,8,9],[2,5,7,8,9,10,11],[2,5,7,8,10],[2,5,7,8,11],[2,5,7,9,10],[2,5,7,9,11],[2,5,7,10,11],[2,5,8],[2,5,8,9,10],[2,5,8,9,11],[2,5,8,10,11],[2,5,9],[2,5,9,10,11],[2,5,10],[2,5,11],[2,6,7],[2,6,7,8,9],[2,6,7,8,9,10,11],[2,6,7,8,10],[2,6,7,8,11],[2,6,7,9,10],[2,6,7,9,11],[2,6,7,10,11],[2,6,8],[2,6,8,9,10],[2,6,8,9,11],[2,6,8,10,11],[2,6,9],[2,6,9,10,11],[2,6,10],[2,6,11],[2,7,8],[2,7,8,9,10],[2,7,8,9,11],[2,7,8,10,11],[2,7,9],[2,7,9,10,11],[2,7,10],[2,7,11],[2,8,9],[2,8,9,10,11],[2,8,10],[2,8,11],[2,9,10],[2,9,11],[2,10,11],[3],[3,4,5],[3,4,5,6,7],[3,4,5,6,7,8,9],[3,4,5,6,7,8,9,10,11],[3,4,5,6,7,8,10],[3,4,5,6,7,8,11],[3,4,5,6,7,9,10],[3,4,5,6,7,9,11],[3,4,5,6,7,10,11],[3,4,5,6,8],[3,4,5,6,8,9,10],[3,4,5,6,8,9,11],[3,4,5,6,8,10,11],[3,4,5,6,9],[3,4,5,6,9,10,11],[3,4,5,6,10],[3,4,5,6,11],[3,4,5,7,8],[3,4,5,7,8,9,10],[3,4,5,7,8,9,11],[3,4,5,7,8,10,11],[3,4,5,7,9],[3,4,5,7,9,10,11],[3,4,5,7,10],[3,4,5,7,11],[3,4,5,8,9],[3,4,5,8,9,10,11],[3,4,5,8,10],[3,4,5,8,11],[3,4,5,9,10],[3,4,5,9,11],[3,4,5,10,11],[3,4,6],[3,4,6,7,8],[3,4,6,7,8,9,10],[3,4,6,7,8,9,11],[3,4,6,7,8,10,11],[3,4,6,7,9],[3,4,6,7,9,10,11],[3,4,6,7,10],[3,4,6,7,11],[3,4,6,8,9],[3,4,6,8,9,10,11],[3,4,6,8,10],[3,4,6,8,11],[3,4,6,9,10],[3,4,6,9,11],[3,4,6,10,11],[3,4,7],[3,4,7,8,9],[3,4,7,8,9,10,11],[3,4,7,8,10],[3,4,7,8,11],[3,4,7,9,10],[3,4,7,9,11],[3,4,7,10,11],[3,4,8],[3,4,8,9,10],[3,4,8,9,11],[3,4,8,10,11],[3,4,9],[3,4,9,10,11],[3,4,10],[3,4,11],[3,5,6],[3,5,6,7,8],[3,5,6,7,8,9,10],[3,5,6,7,8,9,11],[3,5,6,7,8,10,11],[3,5,6,7,9],[3,5,6,7,9,10,11],[3,5,6,7,10],[3,5,6,7,11],[3,5,6,8,9],[3,5,6,8,9,10,11],[3,5,6,8,10],[3,5,6,8,11],[3,5,6,9,10],[3,5,6,9,11],[3,5,6,10,11],[3,5,7],[3,5,7,8,9],[3,5,7,8,9,10,11],[3,5,7,8,10],[3,5,7,8,11],[3,5,7,9,10],[3,5,7,9,11],[3,5,7,10,11],[3,5,8],[3,5,8,9,10],[3,5,8,9,11],[3,5,8,10,11],[3,5,9],[3,5,9,10,11],[3,5,10],[3,5,11],[3,6,7],[3,6,7,8,9],[3,6,7,8,9,10,11],[3,6,7,8,10],[3,6,7,8,11],[3,6,7,9,10],[3,6,7,9,11],[3,6,7,10,11],[3,6,8],[3,6,8,9,10],[3,6,8,9,11],[3,6,8,10,11],[3,6,9],[3,6,9,10,11],[3,6,10],[3,6,11],[3,7,8],[3,7,8,9,10],[3,7,8,9,11],[3,7,8,10,11],[3,7,9],[3,7,9,10,11],[3,7,10],[3,7,11],[3,8,9],[3,8,9,10,11],[3,8,10],[3,8,11],[3,9,10],[3,9,11],[3,10,11],[4],[4,5,6],[4,5,6,7,8],[4,5,6,7,8,9,10],[4,5,6,7,8,9,11],[4,5,6,7,8,10,11],[4,5,6,7,9],[4,5,6,7,9,10,11],[4,5,6,7,10],[4,5,6,7,11],[4,5,6,8,9],[4,5,6,8,9,10,11],[4,5,6,8,10],[4,5,6,8,11],[4,5,6,9,10],[4,5,6,9,11],[4,5,6,10,11],[4,5,7],[4,5,7,8,9],[4,5,7,8,9,10,11],[4,5,7,8,10],[4,5,7,8,11],[4,5,7,9,10],[4,5,7,9,11],[4,5,7,10,11],[4,5,8],[4,5,8,9,10],[4,5,8,9,11],[4,5,8,10,11],[4,5,9],[4,5,9,10,11],[4,5,10],[4,5,11],[4,6,7],[4,6,7,8,9],[4,6,7,8,9,10,11],[4,6,7,8,10],[4,6,7,8,11],[4,6,7,9,10],[4,6,7,9,11],[4,6,7,10,11],[4,6,8],[4,6,8,9,10],[4,6,8,9,11],[4,6,8,10,11],[4,6,9],[4,6,9,10,11],[4,6,10],[4,6,11],[4,7,8],[4,7,8,9,10],[4,7,8,9,11],[4,7,8,10,11],[4,7,9],[4,7,9,10,11],[4,7,10],[4,7,11],[4,8,9],[4,8,9,10,11],[4,8,10],[4,8,11],[4,9,10],[4,9,11],[4,10,11],[5],[5,6,7],[5,6,7,8,9],[5,6,7,8,9,10,11],[5,6,7,8,10],[5,6,7,8,11],[5,6,7,9,10],[5,6,7,9,11],[5,6,7,10,11],[5,6,8],[5,6,8,9,10],[5,6,8,9,11],[5,6,8,10,11],[5,6,9],[5,6,9,10,11],[5,6,10],[5,6,11],[5,7,8],[5,7,8,9,10],[5,7,8,9,11],[5,7,8,10,11],[5,7,9],[5,7,9,10,11],[5,7,10],[5,7,11],[5,8,9],[5,8,9,10,11],[5,8,10],[5,8,11],[5,9,10],[5,9,11],[5,10,11],[6],[6,7,8],[6,7,8,9,10],[6,7,8,9,11],[6,7,8,10,11],[6,7,9],[6,7,9,10,11],[6,7,10],[6,7,11],[6,8,9],[6,8,9,10,11],[6,8,10],[6,8,11],[6,9,10],[6,9,11],[6,10,11],[7],[7,8,9],[7,8,9,10,11],[7,8,10],[7,8,11],[7,9,10],[7,9,11],[7,10,11],[8],[8,9,10],[8,9,11],[8,10,11],[9],[9,10,11],[10],[11]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[0,255],"A_d":1530,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1],[2,1]],"CZ":[]},"rotation_powers":[[0,5],[64,7],[128,7],[255,3],[510,3],[892,3],[956,3],[1017,3],[1018,3],[1019,3],[1020,7],[1116,3],[1180,3],[1209,3],[1210,3],[1211,3]]},"metrics":{"gamma":8.732450113043713,"gamma_t":8.31741261376487,"gamma_rho":null,"gamma_claim":8.732450113043713,"gamma_t_claim":8.31741261376487,"gamma_rho_claim":null,"v_ex":null,"rate":0.0023510971786833857,"t_count":4,"poly_degree":2,"effective_width":3},"provenance":{"regimes":["borrowed-identity search: two-group"],"discovery":"pre-existing","strongest_claim":"found by the two-group borrowed-identity search of Singh, Gidney and Jones (arXiv:2606.28518; borrowed_identities/): an identity circuit symmetric within its output and check blocks, with its output-only gates removed; a verified witness, not a maximum","sources":[{"regime":"borrowed-identity search: two-group","file":"borrowed_identities/circuits/unified_l3.json","label":"ls-0640","N":12,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at 4627014, outputs/large_sweep.csv line 640","provenance":"unified search l=3, two-group s_total=2,s_O=2"}]},"references":[{"key":"singh2026borrowed","short":"Singh et al. (2026)","full":"S. Singh, C. Gidney, and C. Jones, \"Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search,\" arXiv:2606.28518 (2026).","url":"https://arxiv.org/abs/2606.28518","date":"2026-06-26"}]}