Best Known (112, 112+∞, s)-Nets in Base 2
(112, 112+∞, 57)-Net over F2 — Constructive and digital
Digital (112, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (112, 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
(112, 112+∞, 72)-Net over F2 — Digital
Digital (112, m, 72)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (112, 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
(112, 112+∞, 122)-Net in Base 2 — Upper bound on s
There is no (112, m, 123)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (112, 851, 123)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2851, 123, S2, 7, 739), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 3 993998 952924 100522 344655 894010 924650 217856 103768 335881 512642 475885 719991 313431 268628 022225 644736 294611 430738 206263 306113 405191 434472 793415 257097 202332 429545 141134 722818 360171 143086 282141 025359 354229 236532 324197 114103 740371 772216 307448 774588 011465 958351 699968 / 185 > 2851 [i]
- extracting embedded OOA [i] would yield OOA(2851, 123, S2, 7, 739), but