Best Known (51, 51+∞, s)-Nets in Base 2
(51, 51+∞, 36)-Net over F2 — Constructive and digital
Digital (51, m, 36)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
(51, 51+∞, 40)-Net over F2 — Digital
Digital (51, m, 40)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (51, 39)-sequence over F2, using
- t-expansion [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- t-expansion [i] based on digital (50, 39)-sequence over F2, using
(51, 51+∞, 59)-Net in Base 2 — Upper bound on s
There is no (51, m, 60)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (51, 412, 60)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2412, 60, S2, 7, 361), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 073071 518126 219606 313206 405990 769980 901877 085775 985903 676658 621980 536561 317072 380507 793363 938947 133282 046040 910473 251642 146816 / 181 > 2412 [i]
- extracting embedded OOA [i] would yield OOA(2412, 60, S2, 7, 361), but