Best Known (243−122, 243, s)-Nets in Base 2
(243−122, 243, 57)-Net over F2 — Constructive and digital
Digital (121, 243, 57)-net over F2, using
- t-expansion [i] based on digital (110, 243, 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
(243−122, 243, 80)-Net over F2 — Digital
Digital (121, 243, 80)-net over F2, using
- net from sequence [i] based on digital (121, 79)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 121 and N(F) ≥ 80, using
(243−122, 243, 255)-Net over F2 — Upper bound on s (digital)
There is no digital (121, 243, 256)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2243, 256, F2, 122) (dual of [256, 13, 123]-code), but
- residual code [i] would yield OA(2121, 133, S2, 61), but
- 1 times truncation [i] would yield OA(2120, 132, S2, 60), but
- the linear programming bound shows that M ≥ 16503 694795 665515 477973 668460 440758 255616 / 10323 > 2120 [i]
- 1 times truncation [i] would yield OA(2120, 132, S2, 60), but
- residual code [i] would yield OA(2121, 133, S2, 61), but
(243−122, 243, 258)-Net in Base 2 — Upper bound on s
There is no (121, 243, 259)-net in base 2, because
- 14 times m-reduction [i] would yield (121, 229, 259)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2229, 259, S2, 108), but
- the linear programming bound shows that M ≥ 17498 495097 789040 453213 386774 622994 352540 479460 641070 835741 390745 701581 780259 176448 / 19 797109 545047 > 2229 [i]
- extracting embedded orthogonal array [i] would yield OA(2229, 259, S2, 108), but