Best Known (247−130, 247, s)-Nets in Base 2
(247−130, 247, 57)-Net over F2 — Constructive and digital
Digital (117, 247, 57)-net over F2, using
- t-expansion [i] based on digital (110, 247, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-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 8 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 (110, 56)-sequence over F2, using
(247−130, 247, 73)-Net over F2 — Digital
Digital (117, 247, 73)-net over F2, using
- t-expansion [i] based on digital (114, 247, 73)-net over F2, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 114 and N(F) ≥ 73, using
- net from sequence [i] based on digital (114, 72)-sequence over F2, using
(247−130, 247, 246)-Net in Base 2 — Upper bound on s
There is no (117, 247, 247)-net in base 2, because
- 6 times m-reduction [i] would yield (117, 241, 247)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2241, 247, S2, 124), but
- adding a parity check bit [i] would yield OA(2242, 248, S2, 125), but
- the (dual) Plotkin bound shows that M ≥ 452 312848 583266 388373 324160 190187 140051 835877 600158 453279 131187 530910 662656 / 63 > 2242 [i]
- adding a parity check bit [i] would yield OA(2242, 248, S2, 125), but
- extracting embedded orthogonal array [i] would yield OA(2241, 247, S2, 124), but