Best Known (217−37, 217, s)-Nets in Base 2
(217−37, 217, 380)-Net over F2 — Constructive and digital
Digital (180, 217, 380)-net over F2, using
- 22 times duplication [i] based on digital (178, 215, 380)-net over F2, using
- t-expansion [i] based on digital (177, 215, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 43, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 43, 76)-net over F32, using
- t-expansion [i] based on digital (177, 215, 380)-net over F2, using
(217−37, 217, 1131)-Net over F2 — Digital
Digital (180, 217, 1131)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 1131, F2, 3, 37) (dual of [(1131, 3), 3176, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2217, 1365, F2, 3, 37) (dual of [(1365, 3), 3878, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2217, 4095, F2, 37) (dual of [4095, 3878, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 4096, F2, 37) (dual of [4096, 3879, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(2217, 4096, F2, 37) (dual of [4096, 3879, 38]-code), using
- OOA 3-folding [i] based on linear OA(2217, 4095, F2, 37) (dual of [4095, 3878, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2217, 1365, F2, 3, 37) (dual of [(1365, 3), 3878, 38]-NRT-code), using
(217−37, 217, 30911)-Net in Base 2 — Upper bound on s
There is no (180, 217, 30912)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 216, 30912)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 105352 038017 248692 205640 998139 341016 337841 078312 755520 208231 904925 > 2216 [i]