Best Known (76, 76+83, s)-Nets in Base 2
(76, 76+83, 50)-Net over F2 — Constructive and digital
Digital (76, 159, 50)-net over F2, using
- t-expansion [i] based on digital (75, 159, 50)-net over F2, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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 (75, 49)-sequence over F2, using
(76, 76+83, 161)-Net over F2 — Upper bound on s (digital)
There is no digital (76, 159, 162)-net over F2, because
- 3 times m-reduction [i] would yield digital (76, 156, 162)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2156, 162, F2, 80) (dual of [162, 6, 81]-code), but
(76, 76+83, 163)-Net in Base 2 — Upper bound on s
There is no (76, 159, 164)-net in base 2, because
- 1 times m-reduction [i] would yield (76, 158, 164)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2158, 164, S2, 82), but
- adding a parity check bit [i] would yield OA(2159, 165, S2, 83), but
- the (dual) Plotkin bound shows that M ≥ 5 846006 549323 611672 814739 330865 132078 623730 171904 / 7 > 2159 [i]
- adding a parity check bit [i] would yield OA(2159, 165, S2, 83), but
- extracting embedded orthogonal array [i] would yield OA(2158, 164, S2, 82), but