Best Known (187, ∞, s)-Nets in Base 2
(187, ∞, 66)-Net over F2 — Constructive and digital
Digital (187, m, 66)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (187, 65)-sequence over F2, using
- t-expansion [i] based on digital (163, 65)-sequence over F2, using
- base reduction for sequences [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- base reduction for sequences [i] based on digital (49, 65)-sequence over F4, using
- t-expansion [i] based on digital (163, 65)-sequence over F2, using
(187, ∞, 88)-Net over F2 — Digital
Digital (187, m, 88)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (187, 87)-sequence over F2, using
- base reduction for sequences [i] based on digital (50, 87)-sequence over F4, using
- s-reduction based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- s-reduction based on digital (50, 90)-sequence over F4, using
- base reduction for sequences [i] based on digital (50, 87)-sequence over F4, using
(187, ∞, 197)-Net in Base 2 — Upper bound on s
There is no (187, m, 198)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (187, 1772, 198)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(21772, 198, S2, 9, 1585), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 219 852983 669858 992493 497399 002382 040233 863291 354670 876179 511397 454369 068351 867414 184849 843721 261463 131950 099448 522568 908106 724341 604079 103267 077493 242770 201313 661381 871282 557771 689872 266248 436609 918524 812985 828270 254790 331265 266777 280905 786987 870112 342505 687428 580775 596609 764664 018300 301878 944055 854259 532395 836688 702677 409919 772830 887180 516931 663498 957515 079876 589328 539165 589817 588721 359468 484541 226571 617175 017635 108169 003970 456766 584884 808456 492835 844337 812223 703535 856734 576482 483051 637091 453890 967728 192619 626094 433643 949400 784896 / 793 > 21772 [i]
- extracting embedded OOA [i] would yield OOA(21772, 198, S2, 9, 1585), but