Best Known (169−84, 169, s)-Nets in Base 2
(169−84, 169, 52)-Net over F2 — Constructive and digital
Digital (85, 169, 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]
(169−84, 169, 57)-Net over F2 — Digital
Digital (85, 169, 57)-net over F2, using
- t-expansion [i] based on digital (83, 169, 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
(169−84, 169, 183)-Net over F2 — Upper bound on s (digital)
There is no digital (85, 169, 184)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2169, 184, F2, 84) (dual of [184, 15, 85]-code), but
- residual code [i] would yield linear OA(285, 99, F2, 42) (dual of [99, 14, 43]-code), but
(169−84, 169, 211)-Net in Base 2 — Upper bound on s
There is no (85, 169, 212)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 748 602885 726626 685565 542852 092854 259812 380644 506020 > 2169 [i]