Best Known (180−93, 180, s)-Nets in Base 2
(180−93, 180, 52)-Net over F2 — Constructive and digital
Digital (87, 180, 52)-net over F2, using
- t-expansion [i] based on digital (85, 180, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-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 3 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 (85, 51)-sequence over F2, using
(180−93, 180, 57)-Net over F2 — Digital
Digital (87, 180, 57)-net over F2, using
- t-expansion [i] based on digital (83, 180, 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
(180−93, 180, 184)-Net over F2 — Upper bound on s (digital)
There is no digital (87, 180, 185)-net over F2, because
- 5 times m-reduction [i] would yield digital (87, 175, 185)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2175, 185, F2, 88) (dual of [185, 10, 89]-code), but
- residual code [i] would yield linear OA(287, 96, F2, 44) (dual of [96, 9, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2175, 185, F2, 88) (dual of [185, 10, 89]-code), but
(180−93, 180, 205)-Net in Base 2 — Upper bound on s
There is no (87, 180, 206)-net in base 2, because
- 1 times m-reduction [i] would yield (87, 179, 206)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 817388 781522 106567 934698 136689 751586 970657 378524 982800 > 2179 [i]