Best Known (151−77, 151, s)-Nets in Base 2
(151−77, 151, 49)-Net over F2 — Constructive and digital
Digital (74, 151, 49)-net over F2, using
- t-expansion [i] based on digital (70, 151, 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
(151−77, 151, 158)-Net over F2 — Upper bound on s (digital)
There is no digital (74, 151, 159)-net over F2, because
- 1 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
(151−77, 151, 180)-Net in Base 2 — Upper bound on s
There is no (74, 151, 181)-net in base 2, because
- 1 times m-reduction [i] would yield (74, 150, 181)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1464 451125 260000 283177 876578 588813 391594 439676 > 2150 [i]