Best Known (155−78, 155, s)-Nets in Base 2
(155−78, 155, 50)-Net over F2 — Constructive and digital
Digital (77, 155, 50)-net over F2, using
- t-expansion [i] based on digital (75, 155, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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 (75, 49)-sequence over F2, using
(155−78, 155, 52)-Net over F2 — Digital
Digital (77, 155, 52)-net over F2, using
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 77 and N(F) ≥ 52, using
(155−78, 155, 166)-Net over F2 — Upper bound on s (digital)
There is no digital (77, 155, 167)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2155, 167, F2, 78) (dual of [167, 12, 79]-code), but
- residual code [i] would yield linear OA(277, 88, F2, 39) (dual of [88, 11, 40]-code), but
- 1 times truncation [i] would yield linear OA(276, 87, F2, 38) (dual of [87, 11, 39]-code), but
- residual code [i] would yield linear OA(277, 88, F2, 39) (dual of [88, 11, 40]-code), but
(155−78, 155, 189)-Net in Base 2 — Upper bound on s
There is no (77, 155, 190)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 48510 058026 588021 893760 866900 254493 737401 944712 > 2155 [i]