Best Known (165−81, 165, s)-Nets in Base 2
(165−81, 165, 51)-Net over F2 — Constructive and digital
Digital (84, 165, 51)-net over F2, using
- t-expansion [i] based on digital (80, 165, 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
(165−81, 165, 57)-Net over F2 — Digital
Digital (84, 165, 57)-net over F2, using
- t-expansion [i] based on digital (83, 165, 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
(165−81, 165, 188)-Net over F2 — Upper bound on s (digital)
There is no digital (84, 165, 189)-net over F2, because
- 1 times m-reduction [i] would yield digital (84, 164, 189)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2164, 189, F2, 80) (dual of [189, 25, 81]-code), but
- residual code [i] would yield linear OA(284, 108, F2, 40) (dual of [108, 24, 41]-code), but
- adding a parity check bit [i] would yield linear OA(285, 109, F2, 41) (dual of [109, 24, 42]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(285, 109, F2, 41) (dual of [109, 24, 42]-code), but
- residual code [i] would yield linear OA(284, 108, F2, 40) (dual of [108, 24, 41]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2164, 189, F2, 80) (dual of [189, 25, 81]-code), but
(165−81, 165, 216)-Net in Base 2 — Upper bound on s
There is no (84, 165, 217)-net in base 2, because
- 1 times m-reduction [i] would yield (84, 164, 217)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 26 161683 912245 684490 492045 623034 356886 512864 978450 > 2164 [i]