Best Known (106, 106+81, s)-Nets in Base 2
(106, 106+81, 56)-Net over F2 — Constructive and digital
Digital (106, 187, 56)-net over F2, using
- t-expansion [i] based on digital (105, 187, 56)-net over F2, using
- net from sequence [i] based on digital (105, 55)-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 7 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 (105, 55)-sequence over F2, using
(106, 106+81, 73)-Net over F2 — Digital
Digital (106, 187, 73)-net over F2, using
(106, 106+81, 296)-Net in Base 2 — Upper bound on s
There is no (106, 187, 297)-net in base 2, because
- 1 times m-reduction [i] would yield (106, 186, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2186, 297, S2, 80), but
- 2 times code embedding in larger space [i] would yield OA(2188, 299, S2, 80), but
- adding a parity check bit [i] would yield OA(2189, 300, S2, 81), but
- the linear programming bound shows that M ≥ 333112 132422 259965 088720 215000 611770 757764 145862 673192 223023 087630 748136 412561 549060 805697 390758 928463 024902 788757 444450 017971 109260 951552 / 284 805581 394495 166949 489186 936966 421059 043171 883994 793835 511370 537717 840519 721875 > 2189 [i]
- adding a parity check bit [i] would yield OA(2189, 300, S2, 81), but
- 2 times code embedding in larger space [i] would yield OA(2188, 299, S2, 80), but
- extracting embedded orthogonal array [i] would yield OA(2186, 297, S2, 80), but