Best Known (154−72, 154, s)-Nets in Base 2
(154−72, 154, 51)-Net over F2 — Constructive and digital
Digital (82, 154, 51)-net over F2, using
- t-expansion [i] based on digital (80, 154, 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
(154−72, 154, 56)-Net over F2 — Digital
Digital (82, 154, 56)-net over F2, using
- t-expansion [i] based on digital (80, 154, 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
(154−72, 154, 216)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 154, 217)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2154, 217, F2, 72) (dual of [217, 63, 73]-code), but
- residual code [i] would yield OA(282, 144, S2, 36), but
- the linear programming bound shows that M ≥ 2893 609949 429043 518839 097243 777165 200630 895720 113073 291264 / 558 874024 262522 794899 259597 123083 > 282 [i]
- residual code [i] would yield OA(282, 144, S2, 36), but
(154−72, 154, 227)-Net in Base 2 — Upper bound on s
There is no (82, 154, 228)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 23531 724905 569682 391212 542040 466419 524063 298256 > 2154 [i]