Best Known (152−71, 152, s)-Nets in Base 2
(152−71, 152, 51)-Net over F2 — Constructive and digital
Digital (81, 152, 51)-net over F2, using
- t-expansion [i] based on digital (80, 152, 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
(152−71, 152, 56)-Net over F2 — Digital
Digital (81, 152, 56)-net over F2, using
- t-expansion [i] based on digital (80, 152, 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
(152−71, 152, 221)-Net over F2 — Upper bound on s (digital)
There is no digital (81, 152, 222)-net over F2, because
- 1 times m-reduction [i] would yield digital (81, 151, 222)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2151, 222, F2, 70) (dual of [222, 71, 71]-code), but
- construction Y1 [i] would yield
- linear OA(2150, 194, F2, 70) (dual of [194, 44, 71]-code), but
- construction Y1 [i] would yield
- linear OA(2149, 178, F2, 70) (dual of [178, 29, 71]-code), but
- adding a parity check bit [i] would yield linear OA(2150, 179, F2, 71) (dual of [179, 29, 72]-code), but
- OA(244, 194, S2, 16), but
- discarding factors would yield OA(244, 173, S2, 16), but
- the Rao or (dual) Hamming bound shows that M ≥ 17 734855 699135 > 244 [i]
- discarding factors would yield OA(244, 173, S2, 16), but
- linear OA(2149, 178, F2, 70) (dual of [178, 29, 71]-code), but
- construction Y1 [i] would yield
- OA(271, 222, S2, 28), but
- discarding factors would yield OA(271, 209, S2, 28), but
- the Rao or (dual) Hamming bound shows that M ≥ 2400 648072 841355 873622 > 271 [i]
- discarding factors would yield OA(271, 209, S2, 28), but
- linear OA(2150, 194, F2, 70) (dual of [194, 44, 71]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2151, 222, F2, 70) (dual of [222, 71, 71]-code), but
(152−71, 152, 228)-Net in Base 2 — Upper bound on s
There is no (81, 152, 229)-net in base 2, because
- 1 times m-reduction [i] would yield (81, 151, 229)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2950 459742 252515 437070 309221 263528 012660 765984 > 2151 [i]