Best Known (53, 53+∞, s)-Nets in Base 2
(53, 53+∞, 36)-Net over F2 — Constructive and digital
Digital (53, m, 36)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (53, 35)-sequence over F2, using
- t-expansion [i] based on digital (51, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (51, 35)-sequence over F2, using
(53, 53+∞, 40)-Net over F2 — Digital
Digital (53, m, 40)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (53, 39)-sequence over F2, using
- t-expansion [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- t-expansion [i] based on digital (50, 39)-sequence over F2, using
(53, 53+∞, 61)-Net in Base 2 — Upper bound on s
There is no (53, m, 62)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (53, 426, 62)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2426, 62, S2, 7, 373), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 33618 620041 214880 172388 271982 734889 904166 799538 931775 973942 064914 952283 357143 210679 359689 674012 692384 430868 432907 018308 426311 663616 / 187 > 2426 [i]
- extracting embedded OOA [i] would yield OOA(2426, 62, S2, 7, 373), but