Best Known (68, 68+166, s)-Nets in Base 2
(68, 68+166, 43)-Net over F2 — Constructive and digital
Digital (68, 234, 43)-net over F2, using
- t-expansion [i] based on digital (59, 234, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(68, 68+166, 49)-Net over F2 — Digital
Digital (68, 234, 49)-net over F2, using
- net from sequence [i] based on digital (68, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 68 and N(F) ≥ 49, using
(68, 68+166, 99)-Net in Base 2 — Upper bound on s
There is no (68, 234, 100)-net in base 2, because
- 41 times m-reduction [i] would yield (68, 193, 100)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2193, 100, S2, 2, 125), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 803469 022129 495137 770981 046170 581301 261101 496891 396417 650688 / 63 > 2193 [i]
- extracting embedded OOA [i] would yield OOA(2193, 100, S2, 2, 125), but