Best Known (91, ∞, s)-Nets in Base 2
(91, ∞, 53)-Net over F2 — Constructive and digital
Digital (91, m, 53)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (91, 52)-sequence over F2, using
- t-expansion [i] based on digital (90, 52)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 4 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (90, 52)-sequence over F2, using
(91, ∞, 57)-Net over F2 — Digital
Digital (91, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (91, 56)-sequence over F2, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
(91, ∞, 100)-Net in Base 2 — Upper bound on s
There is no (91, m, 101)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (91, 698, 101)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2698, 101, S2, 7, 607), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 25 478783 273124 934175 698544 745200 623223 759458 419924 748381 944798 124239 171372 376485 232113 072285 094569 060630 698755 397113 242047 437874 961827 799962 025352 795345 047866 970614 657028 403522 490871 327252 188161 782826 688095 191040 / 19 > 2698 [i]
- extracting embedded OOA [i] would yield OOA(2698, 101, S2, 7, 607), but