Best Known (50, 69, s)-Nets in Base 2
(50, 69, 75)-Net over F2 — Constructive and digital
Digital (50, 69, 75)-net over F2, using
- trace code for nets [i] based on digital (4, 23, 25)-net over F8, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 4 and N(F) ≥ 25, using
- net from sequence [i] based on digital (4, 24)-sequence over F8, using
(50, 69, 104)-Net over F2 — Digital
Digital (50, 69, 104)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(269, 104, F2, 2, 19) (dual of [(104, 2), 139, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(269, 128, F2, 2, 19) (dual of [(128, 2), 187, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(269, 256, F2, 19) (dual of [256, 187, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 2-folding [i] based on linear OA(269, 256, F2, 19) (dual of [256, 187, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(269, 128, F2, 2, 19) (dual of [(128, 2), 187, 20]-NRT-code), using
(50, 69, 767)-Net in Base 2 — Upper bound on s
There is no (50, 69, 768)-net in base 2, because
- 1 times m-reduction [i] would yield (50, 68, 768)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 297 589446 372689 783265 > 268 [i]