Best Known (81, ∞, s)-Nets in Base 2
(81, ∞, 51)-Net over F2 — Constructive and digital
Digital (81, m, 51)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (81, 50)-sequence over F2, using
- t-expansion [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(81, ∞, 56)-Net over F2 — Digital
Digital (81, m, 56)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (81, 55)-sequence over F2, using
- t-expansion [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- t-expansion [i] based on digital (80, 55)-sequence over F2, using
(81, ∞, 90)-Net in Base 2 — Upper bound on s
There is no (81, m, 91)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (81, 628, 91)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2628, 91, S2, 7, 547), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 183789 722145 425344 500926 102280 407243 312349 292894 534274 547257 511097 016699 850935 490423 211819 791837 470938 348628 193272 235646 060458 075934 313298 269455 351014 200522 720112 090153 672782 413006 406670 090240 / 137 > 2628 [i]
- extracting embedded OOA [i] would yield OOA(2628, 91, S2, 7, 547), but