Best Known (90, 181, s)-Nets in Base 2
(90, 181, 53)-Net over F2 — Constructive and digital
Digital (90, 181, 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, 181, 57)-Net over F2 — Digital
Digital (90, 181, 57)-net over F2, using
- t-expansion [i] based on digital (83, 181, 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, 181, 193)-Net over F2 — Upper bound on s (digital)
There is no digital (90, 181, 194)-net over F2, because
- 1 times m-reduction [i] would yield digital (90, 180, 194)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2180, 194, F2, 90) (dual of [194, 14, 91]-code), but
- residual code [i] would yield linear OA(290, 103, F2, 45) (dual of [103, 13, 46]-code), but
- 1 times truncation [i] would yield linear OA(289, 102, F2, 44) (dual of [102, 13, 45]-code), but
- residual code [i] would yield linear OA(290, 103, F2, 45) (dual of [103, 13, 46]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2180, 194, F2, 90) (dual of [194, 14, 91]-code), but
(90, 181, 221)-Net in Base 2 — Upper bound on s
There is no (90, 181, 222)-net in base 2, because
- 1 times m-reduction [i] would yield (90, 180, 222)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 667504 601729 310927 065210 038242 011903 707437 388654 925790 > 2180 [i]