Best Known (113, ∞, s)-Nets in Base 2
(113, ∞, 57)-Net over F2 — Constructive and digital
Digital (113, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (113, 56)-sequence over F2, using
- t-expansion [i] based on digital (110, 56)-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 8 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 (110, 56)-sequence over F2, using
(113, ∞, 72)-Net over F2 — Digital
Digital (113, m, 72)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (113, 71)-sequence over F2, using
- t-expansion [i] based on digital (110, 71)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 110 and N(F) ≥ 72, using
- t-expansion [i] based on digital (110, 71)-sequence over F2, using
(113, ∞, 123)-Net in Base 2 — Upper bound on s
There is no (113, m, 124)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (113, 858, 124)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2858, 124, S2, 7, 745), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1014 776034 715873 720684 741443 386595 231429 787920 740899 294045 678304 850603 383056 116311 499835 541721 706232 086537 349665 158267 971309 685931 982741 617205 773147 828702 700672 249808 672767 119573 437380 797816 007092 467025 269927 062172 028523 267991 930326 776759 329910 401937 327974 776832 / 373 > 2858 [i]
- extracting embedded OOA [i] would yield OOA(2858, 124, S2, 7, 745), but