Best Known (78, 78+78, s)-Nets in Base 2
(78, 78+78, 50)-Net over F2 — Constructive and digital
Digital (78, 156, 50)-net over F2, using
- t-expansion [i] based on digital (75, 156, 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
(78, 78+78, 52)-Net over F2 — Digital
Digital (78, 156, 52)-net over F2, using
- t-expansion [i] based on digital (77, 156, 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
- net from sequence [i] based on digital (77, 51)-sequence over F2, using
(78, 78+78, 171)-Net over F2 — Upper bound on s (digital)
There is no digital (78, 156, 172)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2156, 172, F2, 78) (dual of [172, 16, 79]-code), but
- residual code [i] would yield OA(278, 93, S2, 39), but
- 1 times truncation [i] would yield OA(277, 92, S2, 38), but
- the linear programming bound shows that M ≥ 2818 610548 431507 920827 449344 / 18125 > 277 [i]
- 1 times truncation [i] would yield OA(277, 92, S2, 38), but
- residual code [i] would yield OA(278, 93, S2, 39), but
(78, 78+78, 193)-Net in Base 2 — Upper bound on s
There is no (78, 156, 194)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 93062 167339 979413 870118 395316 788769 311739 857632 > 2156 [i]