Best Known (65, 65+∞, s)-Nets in Base 2
(65, 65+∞, 43)-Net over F2 — Constructive and digital
Digital (65, m, 43)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (65, 42)-sequence over F2, using
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
(65, 65+∞, 48)-Net over F2 — Digital
Digital (65, m, 48)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (65, 47)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 65 and N(F) ≥ 48, using
(65, 65+∞, 74)-Net in Base 2 — Upper bound on s
There is no (65, m, 75)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (65, 441, 75)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2441, 75, S2, 6, 376), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2407 653274 229197 824944 635113 629015 020046 437007 276741 074682 436284 785867 641875 630621 018007 855489 341810 920025 852820 756714 404833 367894 786048 / 377 > 2441 [i]
- extracting embedded OOA [i] would yield OOA(2441, 75, S2, 6, 376), but