Best Known (83, 166, s)-Nets in Base 2
(83, 166, 51)-Net over F2 — Constructive and digital
Digital (83, 166, 51)-net over F2, using
- t-expansion [i] based on digital (80, 166, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 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]
- net from sequence [i] based on digital (80, 50)-sequence over F2, using
(83, 166, 57)-Net over F2 — Digital
Digital (83, 166, 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
(83, 166, 182)-Net over F2 — Upper bound on s (digital)
There is no digital (83, 166, 183)-net over F2, because
- 1 times m-reduction [i] would yield digital (83, 165, 183)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2165, 183, F2, 82) (dual of [183, 18, 83]-code), but
- residual code [i] would yield OA(283, 100, S2, 41), but
- 1 times truncation [i] would yield OA(282, 99, S2, 40), but
- the linear programming bound shows that M ≥ 9903 520314 283042 199192 993792 / 1885 > 282 [i]
- 1 times truncation [i] would yield OA(282, 99, S2, 40), but
- residual code [i] would yield OA(283, 100, S2, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(2165, 183, F2, 82) (dual of [183, 18, 83]-code), but
(83, 166, 207)-Net in Base 2 — Upper bound on s
There is no (83, 166, 208)-net in base 2, because
- 1 times m-reduction [i] would yield (83, 165, 208)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 51 415465 900440 036011 293574 840555 390408 230771 924830 > 2165 [i]