Best Known (86, 86+86, s)-Nets in Base 2
(86, 86+86, 52)-Net over F2 — Constructive and digital
Digital (86, 172, 52)-net over F2, using
- t-expansion [i] based on digital (85, 172, 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
(86, 86+86, 57)-Net over F2 — Digital
Digital (86, 172, 57)-net over F2, using
- t-expansion [i] based on digital (83, 172, 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
(86, 86+86, 186)-Net over F2 — Upper bound on s (digital)
There is no digital (86, 172, 187)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2172, 187, F2, 86) (dual of [187, 15, 87]-code), but
- residual code [i] would yield linear OA(286, 100, F2, 43) (dual of [100, 14, 44]-code), but
- 1 times truncation [i] would yield linear OA(285, 99, F2, 42) (dual of [99, 14, 43]-code), but
- residual code [i] would yield linear OA(286, 100, F2, 43) (dual of [100, 14, 44]-code), but
(86, 86+86, 212)-Net in Base 2 — Upper bound on s
There is no (86, 172, 213)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 6720 091165 515660 008361 204573 055477 378271 372346 654720 > 2172 [i]