Best Known (77−42, 77, s)-Nets in Base 2
(77−42, 77, 24)-Net over F2 — Constructive and digital
Digital (35, 77, 24)-net over F2, using
- t-expansion [i] based on digital (33, 77, 24)-net over F2, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 33 and N(F) ≥ 24, using
- net from sequence [i] based on digital (33, 23)-sequence over F2, using
(77−42, 77, 29)-Net over F2 — Digital
Digital (35, 77, 29)-net over F2, using
- net from sequence [i] based on digital (35, 28)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 35 and N(F) ≥ 29, using
(77−42, 77, 78)-Net over F2 — Upper bound on s (digital)
There is no digital (35, 77, 79)-net over F2, because
- 6 times m-reduction [i] would yield digital (35, 71, 79)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(271, 79, F2, 36) (dual of [79, 8, 37]-code), but
- adding a parity check bit [i] would yield linear OA(272, 80, F2, 37) (dual of [80, 8, 38]-code), but
- “BJV†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(272, 80, F2, 37) (dual of [80, 8, 38]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(271, 79, F2, 36) (dual of [79, 8, 37]-code), but
(77−42, 77, 79)-Net in Base 2 — Upper bound on s
There is no (35, 77, 80)-net in base 2, because
- 2 times m-reduction [i] would yield (35, 75, 80)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(275, 80, S2, 40), but
- adding a parity check bit [i] would yield OA(276, 81, S2, 41), but
- the (dual) Plotkin bound shows that M ≥ 604462 909807 314587 353088 / 7 > 276 [i]
- adding a parity check bit [i] would yield OA(276, 81, S2, 41), but
- extracting embedded orthogonal array [i] would yield OA(275, 80, S2, 40), but