Best Known (167−77, 167, s)-Nets in Base 2
(167−77, 167, 53)-Net over F2 — Constructive and digital
Digital (90, 167, 53)-net over F2, using
- net from sequence [i] based on digital (90, 52)-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 4 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]
(167−77, 167, 57)-Net over F2 — Digital
Digital (90, 167, 57)-net over F2, using
- t-expansion [i] based on digital (83, 167, 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
(167−77, 167, 252)-Net over F2 — Upper bound on s (digital)
There is no digital (90, 167, 253)-net over F2, because
- 1 times m-reduction [i] would yield digital (90, 166, 253)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2166, 253, F2, 76) (dual of [253, 87, 77]-code), but
- construction Y1 [i] would yield
- linear OA(2165, 219, F2, 76) (dual of [219, 54, 77]-code), but
- construction Y1 [i] would yield
- linear OA(2164, 199, F2, 76) (dual of [199, 35, 77]-code), but
- adding a parity check bit [i] would yield linear OA(2165, 200, F2, 77) (dual of [200, 35, 78]-code), but
- OA(254, 219, S2, 20), but
- discarding factors would yield OA(254, 195, S2, 20), but
- the Rao or (dual) Hamming bound shows that M ≥ 18304 094847 646336 > 254 [i]
- discarding factors would yield OA(254, 195, S2, 20), but
- linear OA(2164, 199, F2, 76) (dual of [199, 35, 77]-code), but
- construction Y1 [i] would yield
- linear OA(287, 253, F2, 34) (dual of [253, 166, 35]-code), but
- discarding factors / shortening the dual code would yield linear OA(287, 249, F2, 34) (dual of [249, 162, 35]-code), but
- the Johnson bound shows that N ≤ 5 479817 731706 550581 576069 715195 852406 189711 333670 < 2162 [i]
- discarding factors / shortening the dual code would yield linear OA(287, 249, F2, 34) (dual of [249, 162, 35]-code), but
- linear OA(2165, 219, F2, 76) (dual of [219, 54, 77]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2166, 253, F2, 76) (dual of [253, 87, 77]-code), but
(167−77, 167, 258)-Net in Base 2 — Upper bound on s
There is no (90, 167, 259)-net in base 2, because
- 1 times m-reduction [i] would yield (90, 166, 259)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 104 400402 654226 917376 535645 314416 774009 903913 219008 > 2166 [i]