Best Known (105, 105+∞, s)-Nets in Base 2
(105, 105+∞, 56)-Net over F2 — Constructive and digital
Digital (105, m, 56)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (105, 55)-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 7 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]
(105, 105+∞, 65)-Net over F2 — Digital
Digital (105, m, 65)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (105, 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
(105, 105+∞, 115)-Net in Base 2 — Upper bound on s
There is no (105, m, 116)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (105, 687, 116)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2687, 116, S2, 6, 582), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 390400 711443 043346 240542 217870 009549 396314 282240 782434 884638 035774 632464 576736 467266 248688 239352 267864 502642 219794 477095 888160 987318 329192 966517 502509 319281 832614 256841 564247 522037 544530 477076 672478 796027 265024 / 583 > 2687 [i]
- extracting embedded OOA [i] would yield OOA(2687, 116, S2, 6, 582), but