Best Known (76, 76+69, s)-Nets in Base 2
(76, 76+69, 50)-Net over F2 — Constructive and digital
Digital (76, 145, 50)-net over F2, using
- t-expansion [i] based on digital (75, 145, 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
(76, 76+69, 194)-Net over F2 — Upper bound on s (digital)
There is no digital (76, 145, 195)-net over F2, because
- 1 times m-reduction [i] would yield digital (76, 144, 195)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2144, 195, F2, 68) (dual of [195, 51, 69]-code), but
- residual code [i] would yield OA(276, 126, S2, 34), but
- the linear programming bound shows that M ≥ 38 459620 016436 403553 321591 630765 416829 509898 862592 / 484 232223 153801 738635 105625 > 276 [i]
- residual code [i] would yield OA(276, 126, S2, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(2144, 195, F2, 68) (dual of [195, 51, 69]-code), but
(76, 76+69, 208)-Net in Base 2 — Upper bound on s
There is no (76, 145, 209)-net in base 2, because
- 1 times m-reduction [i] would yield (76, 144, 209)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 23 667493 754429 969896 600665 053969 839275 851400 > 2144 [i]