Best Known (64, 64+∞, s)-Nets in Base 2
(64, 64+∞, 43)-Net over F2 — Constructive and digital
Digital (64, m, 43)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (64, 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
(64, 64+∞, 44)-Net over F2 — Digital
Digital (64, m, 44)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (64, 43)-sequence over F2, using
- t-expansion [i] based on digital (62, 43)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 62 and N(F) ≥ 44, using
- t-expansion [i] based on digital (62, 43)-sequence over F2, using
(64, 64+∞, 73)-Net in Base 2 — Upper bound on s
There is no (64, m, 74)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (64, 435, 74)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2435, 74, S2, 6, 371), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3 194115 487627 178718 234333 140132 832426 358775 510832 940694 596392 476868 043746 592080 717742 050515 831968 793968 194675 027536 914746 978229 813248 / 31 > 2435 [i]
- extracting embedded OOA [i] would yield OOA(2435, 74, S2, 6, 371), but