Best Known (82, 82+81, s)-Nets in Base 2
(82, 82+81, 51)-Net over F2 — Constructive and digital
Digital (82, 163, 51)-net over F2, using
- t-expansion [i] based on digital (80, 163, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-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 2 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 (80, 50)-sequence over F2, using
(82, 82+81, 56)-Net over F2 — Digital
Digital (82, 163, 56)-net over F2, using
- t-expansion [i] based on digital (80, 163, 56)-net over F2, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
(82, 82+81, 179)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 163, 180)-net over F2, because
- 1 times m-reduction [i] would yield digital (82, 162, 180)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2162, 180, F2, 80) (dual of [180, 18, 81]-code), but
- residual code [i] would yield OA(282, 99, S2, 40), but
- the linear programming bound shows that M ≥ 9903 520314 283042 199192 993792 / 1885 > 282 [i]
- residual code [i] would yield OA(282, 99, S2, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(2162, 180, F2, 80) (dual of [180, 18, 81]-code), but
(82, 82+81, 207)-Net in Base 2 — Upper bound on s
There is no (82, 163, 208)-net in base 2, because
- 1 times m-reduction [i] would yield (82, 162, 208)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6 610679 273520 226065 990670 711568 638568 512392 702270 > 2162 [i]