Best Known (158−87, 158, s)-Nets in Base 2
(158−87, 158, 49)-Net over F2 — Constructive and digital
Digital (71, 158, 49)-net over F2, using
- t-expansion [i] based on digital (70, 158, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [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 (70, 48)-sequence over F2, using
(158−87, 158, 151)-Net over F2 — Upper bound on s (digital)
There is no digital (71, 158, 152)-net over F2, because
- 15 times m-reduction [i] would yield digital (71, 143, 152)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2143, 152, F2, 72) (dual of [152, 9, 73]-code), but
- residual code [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
- residual code [i] would yield linear OA(271, 79, F2, 36) (dual of [79, 8, 37]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2143, 152, F2, 72) (dual of [152, 9, 73]-code), but
(158−87, 158, 154)-Net in Base 2 — Upper bound on s
There is no (71, 158, 155)-net in base 2, because
- 9 times m-reduction [i] would yield (71, 149, 155)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2149, 155, S2, 78), but
- the (dual) Plotkin bound shows that M ≥ 68507 889249 886074 290797 726533 575766 546371 837952 / 79 > 2149 [i]
- extracting embedded orthogonal array [i] would yield OA(2149, 155, S2, 78), but