Best Known (212−96, 212, s)-Nets in Base 2
(212−96, 212, 57)-Net over F2 — Constructive and digital
Digital (116, 212, 57)-net over F2, using
- t-expansion [i] based on digital (110, 212, 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
(212−96, 212, 73)-Net over F2 — Digital
Digital (116, 212, 73)-net over F2, using
- t-expansion [i] based on digital (114, 212, 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
(212−96, 212, 291)-Net in Base 2 — Upper bound on s
There is no (116, 212, 292)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2212, 292, S2, 96), but
- 7 times code embedding in larger space [i] would yield OA(2219, 299, S2, 96), but
- adding a parity check bit [i] would yield OA(2220, 300, S2, 97), but
- the linear programming bound shows that M ≥ 2 707506 179671 220537 436761 186277 555172 982275 381427 765331 317661 637213 928123 767152 572244 071765 637582 553088 / 1 434637 915544 533146 412054 075529 090625 > 2220 [i]
- adding a parity check bit [i] would yield OA(2220, 300, S2, 97), but
- 7 times code embedding in larger space [i] would yield OA(2219, 299, S2, 96), but