Best Known (256−126, 256, s)-Nets in Base 2
(256−126, 256, 57)-Net over F2 — Constructive and digital
Digital (130, 256, 57)-net over F2, using
- t-expansion [i] based on digital (110, 256, 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
(256−126, 256, 81)-Net over F2 — Digital
Digital (130, 256, 81)-net over F2, using
- t-expansion [i] based on digital (126, 256, 81)-net over F2, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- net from sequence [i] based on digital (126, 80)-sequence over F2, using
(256−126, 256, 277)-Net over F2 — Upper bound on s (digital)
There is no digital (130, 256, 278)-net over F2, because
- 2 times m-reduction [i] would yield digital (130, 254, 278)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2254, 278, F2, 124) (dual of [278, 24, 125]-code), but
- residual code [i] would yield linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- adding a parity check bit [i] would yield linear OA(2131, 154, F2, 63) (dual of [154, 23, 64]-code), but
- residual code [i] would yield linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2254, 278, F2, 124) (dual of [278, 24, 125]-code), but
(256−126, 256, 286)-Net in Base 2 — Upper bound on s
There is no (130, 256, 287)-net in base 2, because
- 18 times m-reduction [i] would yield (130, 238, 287)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2238, 287, S2, 108), but
- the linear programming bound shows that M ≥ 2 133520 632200 595168 112459 587583 515002 010526 895398 363193 388655 973626 286863 730797 742512 930816 / 4 628313 007677 734375 > 2238 [i]
- extracting embedded orthogonal array [i] would yield OA(2238, 287, S2, 108), but