Best Known (157−76, 157, s)-Nets in Base 2
(157−76, 157, 51)-Net over F2 — Constructive and digital
Digital (81, 157, 51)-net over F2, using
- t-expansion [i] based on digital (80, 157, 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
(157−76, 157, 56)-Net over F2 — Digital
Digital (81, 157, 56)-net over F2, using
- t-expansion [i] based on digital (80, 157, 56)-net over F2, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
(157−76, 157, 177)-Net over F2 — Upper bound on s (digital)
There is no digital (81, 157, 178)-net over F2, because
- 2 times m-reduction [i] would yield digital (81, 155, 178)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- adding a parity check bit [i] would yield linear OA(2156, 179, F2, 75) (dual of [179, 23, 76]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
(157−76, 157, 211)-Net in Base 2 — Upper bound on s
There is no (81, 157, 212)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 187748 483669 408373 917944 476807 890604 876691 210200 > 2157 [i]