Best Known (165−117, 165, s)-Nets in Base 2
(165−117, 165, 35)-Net over F2 — Constructive and digital
Digital (48, 165, 35)-net over F2, using
- net from sequence [i] based on digital (48, 34)-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 2 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]
(165−117, 165, 36)-Net over F2 — Digital
Digital (48, 165, 36)-net over F2, using
- t-expansion [i] based on digital (47, 165, 36)-net over F2, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 47 and N(F) ≥ 36, using
- net from sequence [i] based on digital (47, 35)-sequence over F2, using
(165−117, 165, 72)-Net in Base 2 — Upper bound on s
There is no (48, 165, 73)-net in base 2, because
- 26 times m-reduction [i] would yield (48, 139, 73)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2139, 73, S2, 2, 91), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 16 725558 898897 967356 151788 704486 271129 485312 / 23 > 2139 [i]
- extracting embedded OOA [i] would yield OOA(2139, 73, S2, 2, 91), but