Best Known (82, ∞, s)-Nets in Base 2
(82, ∞, 51)-Net over F2 — Constructive and digital
Digital (82, m, 51)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (82, 50)-sequence over F2, using
- t-expansion [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(82, ∞, 56)-Net over F2 — Digital
Digital (82, m, 56)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (82, 55)-sequence over F2, using
- t-expansion [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- t-expansion [i] based on digital (80, 55)-sequence over F2, using
(82, ∞, 91)-Net in Base 2 — Upper bound on s
There is no (82, m, 92)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (82, 635, 92)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2635, 92, S2, 7, 553), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 46 765016 330627 500991 072008 958427 986080 155592 199297 739288 436723 308467 570463 888943 211685 606316 003056 847731 083673 129342 674934 559223 394098 239482 089778 527152 695429 582945 412192 715738 349460 470527 688704 / 277 > 2635 [i]
- extracting embedded OOA [i] would yield OOA(2635, 92, S2, 7, 553), but