Best Known (117, 117+∞, s)-Nets in Base 2
(117, 117+∞, 57)-Net over F2 — Constructive and digital
Digital (117, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (117, 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
(117, 117+∞, 73)-Net over F2 — Digital
Digital (117, m, 73)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (117, 72)-sequence over F2, using
- t-expansion [i] based on digital (114, 72)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 114 and N(F) ≥ 73, using
- t-expansion [i] based on digital (114, 72)-sequence over F2, using
(117, 117+∞, 127)-Net in Base 2 — Upper bound on s
There is no (117, m, 128)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (117, 886, 128)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2886, 128, S2, 7, 769), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 264147 265567 832623 176169 892458 258303 259423 663018 060761 063980 354513 336951 278362 429737 208627 943828 593947 337197 496628 564339 441173 779751 342768 625269 489231 469788 454193 341999 502542 084365 758838 213220 526512 116454 105594 202074 014146 375780 869419 198449 383518 238244 769290 448868 999168 / 385 > 2886 [i]
- extracting embedded OOA [i] would yield OOA(2886, 128, S2, 7, 769), but