Best Known (238−106, 238, s)-Nets in Base 2
(238−106, 238, 66)-Net over F2 — Constructive and digital
Digital (132, 238, 66)-net over F2, using
- 2 times m-reduction [i] based on digital (132, 240, 66)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (39, 93, 33)-net over F2, using
- net from sequence [i] based on digital (39, 32)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 39 and N(F) ≥ 33, using
- net from sequence [i] based on digital (39, 32)-sequence over F2, using
- digital (39, 147, 33)-net over F2, using
- net from sequence [i] based on digital (39, 32)-sequence over F2 (see above)
- digital (39, 93, 33)-net over F2, using
- (u, u+v)-construction [i] based on
(238−106, 238, 84)-Net over F2 — Digital
Digital (132, 238, 84)-net over F2, using
(238−106, 238, 334)-Net over F2 — Upper bound on s (digital)
There is no digital (132, 238, 335)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2238, 335, F2, 106) (dual of [335, 97, 107]-code), but
- construction Y1 [i] would yield
- OA(2237, 299, S2, 106), but
- the linear programming bound shows that M ≥ 2 372623 864838 720824 144751 616230 912568 605364 469958 081012 522169 374458 839809 772727 298051 538944 / 10 001354 293176 388857 > 2237 [i]
- OA(297, 335, S2, 36), but
- discarding factors would yield OA(297, 324, S2, 36), but
- the Rao or (dual) Hamming bound shows that M ≥ 158757 007614 184387 760434 595956 > 297 [i]
- discarding factors would yield OA(297, 324, S2, 36), but
- OA(2237, 299, S2, 106), but
- construction Y1 [i] would yield
(238−106, 238, 389)-Net in Base 2 — Upper bound on s
There is no (132, 238, 390)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 459926 634048 295129 196262 794727 023247 399016 223217 871697 382212 648168 333864 > 2238 [i]