Best Known (112−59, 112, s)-Nets in Base 2
(112−59, 112, 36)-Net over F2 — Constructive and digital
Digital (53, 112, 36)-net over F2, using
- t-expansion [i] based on digital (51, 112, 36)-net over F2, using
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (51, 35)-sequence over F2, using
(112−59, 112, 40)-Net over F2 — Digital
Digital (53, 112, 40)-net over F2, using
- t-expansion [i] based on digital (50, 112, 40)-net over F2, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 50 and N(F) ≥ 40, using
- net from sequence [i] based on digital (50, 39)-sequence over F2, using
(112−59, 112, 116)-Net in Base 2 — Upper bound on s
There is no (53, 112, 117)-net in base 2, because
- 1 times m-reduction [i] would yield (53, 111, 117)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2111, 117, S2, 58), but
- adding a parity check bit [i] would yield OA(2112, 118, S2, 59), but
- the (dual) Plotkin bound shows that M ≥ 83076 749736 557242 056487 941267 521536 / 15 > 2112 [i]
- adding a parity check bit [i] would yield OA(2112, 118, S2, 59), but
- extracting embedded orthogonal array [i] would yield OA(2111, 117, S2, 58), but