Best Known (90, 178, s)-Nets in Base 2
(90, 178, 53)-Net over F2 — Constructive and digital
Digital (90, 178, 53)-net over F2, using
- net from sequence [i] based on digital (90, 52)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 4 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(90, 178, 57)-Net over F2 — Digital
Digital (90, 178, 57)-net over F2, using
- t-expansion [i] based on digital (83, 178, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(90, 178, 194)-Net over F2 — Upper bound on s (digital)
There is no digital (90, 178, 195)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2178, 195, F2, 88) (dual of [195, 17, 89]-code), but
- residual code [i] would yield OA(290, 106, S2, 44), but
- the linear programming bound shows that M ≥ 60 158934 149112 339838 997840 789504 / 47817 > 290 [i]
- residual code [i] would yield OA(290, 106, S2, 44), but
(90, 178, 225)-Net in Base 2 — Upper bound on s
There is no (90, 178, 226)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 411403 522515 537938 479658 891674 153102 121054 379890 927568 > 2178 [i]