Best Known (51, 51+138, s)-Nets in Base 2
(51, 51+138, 36)-Net over F2 — Constructive and digital
Digital (51, 189, 36)-net over F2, using
- net from sequence [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]
(51, 51+138, 40)-Net over F2 — Digital
Digital (51, 189, 40)-net over F2, using
- t-expansion [i] based on digital (50, 189, 40)-net over F2, using
- net from sequence [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
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(51, 51+138, 76)-Net in Base 2 — Upper bound on s
There is no (51, 189, 77)-net in base 2, because
- 41 times m-reduction [i] would yield (51, 148, 77)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2148, 77, S2, 2, 97), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2854 495385 411919 762116 571938 898990 272765 493248 / 7 > 2148 [i]
- extracting embedded OOA [i] would yield OOA(2148, 77, S2, 2, 97), but