Best Known (158−84, 158, s)-Nets in Base 2
(158−84, 158, 49)-Net over F2 — Constructive and digital
Digital (74, 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
(158−84, 158, 158)-Net over F2 — Upper bound on s (digital)
There is no digital (74, 158, 159)-net over F2, because
- 8 times m-reduction [i] would yield digital (74, 150, 159)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2150, 159, F2, 76) (dual of [159, 9, 77]-code), but
- residual code [i] would yield linear OA(274, 82, F2, 38) (dual of [82, 8, 39]-code), but
- adding a parity check bit [i] would yield linear OA(275, 83, F2, 39) (dual of [83, 8, 40]-code), but
- “DHM†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(275, 83, F2, 39) (dual of [83, 8, 40]-code), but
- residual code [i] would yield linear OA(274, 82, F2, 38) (dual of [82, 8, 39]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2150, 159, F2, 76) (dual of [159, 9, 77]-code), but
(158−84, 158, 159)-Net in Base 2 — Upper bound on s
There is no (74, 158, 160)-net in base 2, because
- 4 times m-reduction [i] would yield (74, 154, 160)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2154, 160, S2, 80), but
- adding a parity check bit [i] would yield OA(2155, 161, S2, 81), but
- the (dual) Plotkin bound shows that M ≥ 2 192252 455996 354377 305527 249074 424529 483898 814464 / 41 > 2155 [i]
- adding a parity check bit [i] would yield OA(2155, 161, S2, 81), but
- extracting embedded orthogonal array [i] would yield OA(2154, 160, S2, 80), but