Best Known (164−81, 164, s)-Nets in Base 2
(164−81, 164, 51)-Net over F2 — Constructive and digital
Digital (83, 164, 51)-net over F2, using
- t-expansion [i] based on digital (80, 164, 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
(164−81, 164, 57)-Net over F2 — Digital
Digital (83, 164, 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
(164−81, 164, 183)-Net over F2 — Upper bound on s (digital)
There is no digital (83, 164, 184)-net over F2, because
- 1 times m-reduction [i] would yield digital (83, 163, 184)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2163, 184, F2, 80) (dual of [184, 21, 81]-code), but
- residual code [i] would yield linear OA(283, 103, F2, 40) (dual of [103, 20, 41]-code), but
- adding a parity check bit [i] would yield linear OA(284, 104, F2, 41) (dual of [104, 20, 42]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(284, 104, F2, 41) (dual of [104, 20, 42]-code), but
- residual code [i] would yield linear OA(283, 103, F2, 40) (dual of [103, 20, 41]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2163, 184, F2, 80) (dual of [184, 21, 81]-code), but
(164−81, 164, 211)-Net in Base 2 — Upper bound on s
There is no (83, 164, 212)-net in base 2, because
- 1 times m-reduction [i] would yield (83, 163, 212)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 12 257184 037477 093310 412480 381587 738329 731901 456140 > 2163 [i]