Best Known (104, ∞, s)-Nets in Base 2
(104, ∞, 55)-Net over F2 — Constructive and digital
Digital (104, m, 55)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (104, 54)-sequence over F2, using
- t-expansion [i] based on digital (100, 54)-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 6 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 (100, 54)-sequence over F2, using
(104, ∞, 65)-Net over F2 — Digital
Digital (104, m, 65)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (104, 64)-sequence over F2, using
- t-expansion [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- t-expansion [i] based on digital (95, 64)-sequence over F2, using
(104, ∞, 114)-Net in Base 2 — Upper bound on s
There is no (104, m, 115)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (104, 681, 115)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2681, 115, S2, 6, 577), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3090 137210 229681 091747 712867 597731 836298 951452 769351 098490 658794 679965 025946 618830 129641 138407 701955 753156 193557 172567 473621 340051 758979 015051 587961 885031 322071 473866 858598 587499 680399 758093 637313 000338 620416 / 289 > 2681 [i]
- extracting embedded OOA [i] would yield OOA(2681, 115, S2, 6, 577), but