Best Known (78−42, 78, s)-Nets in Base 2
(78−42, 78, 24)-Net over F2 — Constructive and digital
Digital (36, 78, 24)-net over F2, using
- t-expansion [i] based on digital (33, 78, 24)-net over F2, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 24, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
(78−42, 78, 30)-Net over F2 — Digital
Digital (36, 78, 30)-net over F2, using
- net from sequence [i] based on digital (36, 29)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 36 and N(F) ≥ 30, using
(78−42, 78, 80)-Net over F2 — Upper bound on s (digital)
There is no digital (36, 78, 81)-net over F2, because
- 2 times m-reduction [i] would yield digital (36, 76, 81)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(276, 81, F2, 40) (dual of [81, 5, 41]-code), but
(78−42, 78, 82)-Net in Base 2 — Upper bound on s
There is no (36, 78, 83)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(278, 83, S2, 42), but
- the (dual) Plotkin bound shows that M ≥ 14 507109 835375 550096 474112 / 43 > 278 [i]