Best Known (181−97, 181, s)-Nets in Base 2
(181−97, 181, 51)-Net over F2 — Constructive and digital
Digital (84, 181, 51)-net over F2, using
- t-expansion [i] based on digital (80, 181, 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
(181−97, 181, 57)-Net over F2 — Digital
Digital (84, 181, 57)-net over F2, using
- t-expansion [i] based on digital (83, 181, 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
(181−97, 181, 178)-Net over F2 — Upper bound on s (digital)
There is no digital (84, 181, 179)-net over F2, because
- 9 times m-reduction [i] would yield digital (84, 172, 179)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2172, 179, F2, 88) (dual of [179, 7, 89]-code), but
(181−97, 181, 179)-Net in Base 2 — Upper bound on s
There is no (84, 181, 180)-net in base 2, because
- 7 times m-reduction [i] would yield (84, 174, 180)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2174, 180, S2, 90), but
- adding a parity check bit [i] would yield OA(2175, 181, S2, 91), but
- the (dual) Plotkin bound shows that M ≥ 1 149371 655649 416643 768760 270362 731887 714054 341637 701632 / 23 > 2175 [i]
- adding a parity check bit [i] would yield OA(2175, 181, S2, 91), but
- extracting embedded orthogonal array [i] would yield OA(2174, 180, S2, 90), but