Best Known (146−35, 146, s)-Nets in Base 2
(146−35, 146, 144)-Net over F2 — Constructive and digital
Digital (111, 146, 144)-net over F2, using
- 4 times m-reduction [i] based on digital (111, 150, 144)-net over F2, using
- trace code for nets [i] based on digital (11, 50, 48)-net over F8, using
- net from sequence [i] based on digital (11, 47)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 11 and N(F) ≥ 48, using
- net from sequence [i] based on digital (11, 47)-sequence over F8, using
- trace code for nets [i] based on digital (11, 50, 48)-net over F8, using
(146−35, 146, 247)-Net over F2 — Digital
Digital (111, 146, 247)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2146, 247, F2, 2, 35) (dual of [(247, 2), 348, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2146, 256, F2, 2, 35) (dual of [(256, 2), 366, 36]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2145, 256, F2, 2, 35) (dual of [(256, 2), 367, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2145, 512, F2, 35) (dual of [512, 367, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 2-folding [i] based on linear OA(2145, 512, F2, 35) (dual of [512, 367, 36]-code), using
- 21 times duplication [i] based on linear OOA(2145, 256, F2, 2, 35) (dual of [(256, 2), 367, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2146, 256, F2, 2, 35) (dual of [(256, 2), 366, 36]-NRT-code), using
(146−35, 146, 2626)-Net in Base 2 — Upper bound on s
There is no (111, 146, 2627)-net in base 2, because
- 1 times m-reduction [i] would yield (111, 145, 2627)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 44 605126 353791 760816 641928 428974 896821 758384 > 2145 [i]