Recall the original basis U = { U0, U1, U2, U3, U4, U5, U6, U7 }. From a symbol ♦wxyz in the table below, select the indicated members of U and construct the subspace whose basis is U′ = { Uw, Ux, Uy, Uz }.
Each diamond quadruple indicates a subspace U′ in which a 3-in-8 is a closed operation, and this applies to any 3-in-8 that is based on that permutation. Each permutation is shared by precisely thirty-two 3-in-8s, which differ in the distribution of positive and negative signs.
| permutation designator | |
|---|---|
| p01 | ♦0123 ♦0145 ♦0167 ♦0246 ♦0257 ♦0347 ♦0356 ♦1247 ♦1256 ♦1346 ♦1357 ♦2345 ♦2367 ♦4567 |
| p02 | ♦0123 ♦0145 ♦0167 ♦0247 ♦0256 ♦0346 ♦0357 ♦1246 ♦1257 ♦1347 ♦1356 ♦2345 ♦2367 ♦4567 |
| p03 | ♦0123 ♦0146 ♦0157 ♦0245 ♦0267 ♦0347 ♦0356 ♦1247 ♦1256 ♦1345 ♦1367 ♦2346 ♦2357 ♦4567 |
| p04 | ♦0123 ♦0146 ♦0157 ♦0247 ♦0256 ♦0345 ♦0367 ♦1245 ♦1267 ♦1347 ♦1356 ♦2346 ♦2357 ♦4567 |
| p05 | ♦0123 ♦0147 ♦0156 ♦0246 ♦0257 ♦0345 ♦0367 ♦1245 ♦1267 ♦1346 ♦1357 ♦2347 ♦2356 ♦4567 |
| p06 | ♦0123 ♦0147 ♦0156 ♦0245 ♦0267 ♦0346 ♦0357 ♦1246 ♦1257 ♦1345 ♦1367 ♦2347 ♦2356 ♦4567 |
| p07 | ♦0124 ♦0135 ♦0167 ♦0236 ♦0257 ♦0347 ♦0456 ♦1237 ♦1256 ♦1346 ♦1457 ♦2345 ♦2467 ♦3567 |
| p08 | ♦0124 ♦0135 ♦0167 ♦0237 ♦0256 ♦0346 ♦0457 ♦1236 ♦1257 ♦1347 ♦1456 ♦2345 ♦2467 ♦3567 |
| p09 | ♦0124 ♦0136 ♦0157 ♦0235 ♦0267 ♦0347 ♦0456 ♦1237 ♦1256 ♦1345 ♦1467 ♦2346 ♦2457 ♦3567 |
| p10 | ♦0124 ♦0136 ♦0157 ♦0237 ♦0256 ♦0345 ♦0467 ♦1235 ♦1267 ♦1347 ♦1456 ♦2346 ♦2457 ♦3567 |
| p11 | ♦0124 ♦0137 ♦0156 ♦0236 ♦0257 ♦0345 ♦0467 ♦1235 ♦1267 ♦1346 ♦1457 ♦2347 ♦2456 ♦3567 |
| p12 | ♦0124 ♦0137 ♦0156 ♦0235 ♦0267 ♦0346 ♦0457 ♦1236 ♦1257 ♦1345 ♦1467 ♦2347 ♦2456 ♦3567 |
| p13 | ♦0125 ♦0134 ♦0167 ♦0237 ♦0246 ♦0356 ♦0457 ♦1236 ♦1247 ♦1357 ♦1456 ♦2345 ♦2567 ♦3467 |
| p14 | ♦0125 ♦0134 ♦0167 ♦0236 ♦0247 ♦0357 ♦0456 ♦1237 ♦1246 ♦1356 ♦1457 ♦2345 ♦2567 ♦3467 |
| p15 | ♦0125 ♦0137 ♦0146 ♦0234 ♦0267 ♦0356 ♦0457 ♦1236 ♦1247 ♦1345 ♦1567 ♦2357 ♦2456 ♦3467 |
| p16 | ♦0125 ♦0137 ♦0146 ♦0236 ♦0247 ♦0345 ♦0567 ♦1234 ♦1267 ♦1356 ♦1457 ♦2357 ♦2456 ♦3467 |
| p17 | ♦0125 ♦0136 ♦0147 ♦0237 ♦0246 ♦0345 ♦0567 ♦1234 ♦1267 ♦1357 ♦1456 ♦2356 ♦2457 ♦3467 |
| p18 | ♦0125 ♦0136 ♦0147 ♦0234 ♦0267 ♦0357 ♦0456 ♦1237 ♦1246 ♦1345 ♦1567 ♦2356 ♦2457 ♦3467 |
| p19 | ♦0126 ♦0137 ♦0145 ♦0234 ♦0257 ♦0356 ♦0467 ♦1235 ♦1247 ♦1346 ♦1567 ♦2367 ♦2456 ♦3457 |
| p20 | ♦0126 ♦0137 ♦0145 ♦0235 ♦0247 ♦0346 ♦0567 ♦1234 ♦1257 ♦1356 ♦1467 ♦2367 ♦2456 ♦3457 |
| p21 | ♦0126 ♦0134 ♦0157 ♦0237 ♦0245 ♦0356 ♦0467 ♦1235 ♦1247 ♦1367 ♦1456 ♦2346 ♦2567 ♦3457 |
| p22 | ♦0126 ♦0134 ♦0157 ♦0235 ♦0247 ♦0367 ♦0456 ♦1237 ♦1245 ♦1356 ♦1467 ♦2346 ♦2567 ♦3457 |
| p23 | ♦0126 ♦0135 ♦0147 ♦0234 ♦0257 ♦0367 ♦0456 ♦1237 ♦1245 ♦1346 ♦1567 ♦2356 ♦2467 ♦3457 |
| p24 | ♦0126 ♦0135 ♦0147 ♦0237 ♦0245 ♦0346 ♦0567 ♦1234 ♦1257 ♦1367 ♦1456 ♦2356 ♦2467 ♦3457 |
| p25 | ♦0127 ♦0136 ♦0145 ♦0235 ♦0246 ♦0347 ♦0567 ♦1234 ♦1256 ♦1357 ♦1467 ♦2367 ♦2457 ♦3456 |
| p26 | ♦0127 ♦0136 ♦0145 ♦0234 ♦0256 ♦0357 ♦0467 ♦1235 ♦1246 ♦1347 ♦1567 ♦2367 ♦2457 ♦3456 |
| p27 | ♦0127 ♦0135 ♦0146 ♦0236 ♦0245 ♦0347 ♦0567 ♦1234 ♦1256 ♦1367 ♦1457 ♦2357 ♦2467 ♦3456 |
| p28 | ♦0127 ♦0135 ♦0146 ♦0234 ♦0256 ♦0367 ♦0457 ♦1236 ♦1245 ♦1347 ♦1567 ♦2357 ♦2467 ♦3456 |
| p29 | ♦0127 ♦0134 ♦0156 ♦0235 ♦0246 ♦0367 ♦0457 ♦1236 ♦1245 ♦1357 ♦1467 ♦2347 ♦2567 ♦3456 |
| p30 | ♦0127 ♦0134 ♦0156 ♦0236 ♦0245 ♦0357 ♦0467 ♦1235 ♦1246 ♦1367 ♦1457 ♦2347 ♦2567 ♦3456 |