Best Known (168−91, 168, s)-Nets in Base 2
(168−91, 168, 50)-Net over F2 — Constructive and digital
Digital (77, 168, 50)-net over F2, using
- t-expansion [i] based on digital (75, 168, 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
(168−91, 168, 52)-Net over F2 — Digital
Digital (77, 168, 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
(168−91, 168, 163)-Net over F2 — Upper bound on s (digital)
There is no digital (77, 168, 164)-net over F2, because
- 11 times m-reduction [i] would yield digital (77, 157, 164)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2157, 164, F2, 80) (dual of [164, 7, 81]-code), but
(168−91, 168, 166)-Net in Base 2 — Upper bound on s
There is no (77, 168, 167)-net in base 2, because
- 7 times m-reduction [i] would yield (77, 161, 167)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2161, 167, S2, 84), but
- the (dual) Plotkin bound shows that M ≥ 280 608314 367533 360295 107487 881526 339773 939048 251392 / 85 > 2161 [i]
- extracting embedded orthogonal array [i] would yield OA(2161, 167, S2, 84), but