Best Known (179−91, 179, s)-Nets in Base 2
(179−91, 179, 52)-Net over F2 — Constructive and digital
Digital (88, 179, 52)-net over F2, using
- t-expansion [i] based on digital (85, 179, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-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 3 places 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 (85, 51)-sequence over F2, using
(179−91, 179, 57)-Net over F2 — Digital
Digital (88, 179, 57)-net over F2, using
- t-expansion [i] based on digital (83, 179, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(179−91, 179, 186)-Net over F2 — Upper bound on s (digital)
There is no digital (88, 179, 187)-net over F2, because
- 3 times m-reduction [i] would yield digital (88, 176, 187)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 187, F2, 88) (dual of [187, 11, 89]-code), but
- residual code [i] would yield linear OA(288, 98, F2, 44) (dual of [98, 10, 45]-code), but
- adding a parity check bit [i] would yield linear OA(289, 99, F2, 45) (dual of [99, 10, 46]-code), but
- residual code [i] would yield linear OA(288, 98, F2, 44) (dual of [98, 10, 45]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2176, 187, F2, 88) (dual of [187, 11, 89]-code), but
(179−91, 179, 213)-Net in Base 2 — Upper bound on s
There is no (88, 179, 214)-net in base 2, because
- 1 times m-reduction [i] would yield (88, 178, 214)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 445286 789851 785348 835623 265536 600827 831362 694287 996860 > 2178 [i]