Best Known (68−18, 68, s)-Nets in Base 2
(68−18, 68, 75)-Net over F2 — Constructive and digital
Digital (50, 68, 75)-net over F2, using
- 1 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (4, 23, 25)-net over F8, using
(68−18, 68, 116)-Net over F2 — Digital
Digital (50, 68, 116)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(268, 116, F2, 2, 18) (dual of [(116, 2), 164, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(268, 127, F2, 2, 18) (dual of [(127, 2), 186, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(268, 254, F2, 18) (dual of [254, 186, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(268, 255, F2, 18) (dual of [255, 187, 19]-code), using
- 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]
- discarding factors / shortening the dual code based on linear OA(268, 255, F2, 18) (dual of [255, 187, 19]-code), using
- OOA 2-folding [i] based on linear OA(268, 254, F2, 18) (dual of [254, 186, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(268, 127, F2, 2, 18) (dual of [(127, 2), 186, 19]-NRT-code), using
(68−18, 68, 767)-Net in Base 2 — Upper bound on s
There is no (50, 68, 768)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 297 589446 372689 783265 > 268 [i]