Best Known (162, 162+35, s)-Nets in Base 2
(162, 162+35, 320)-Net over F2 — Constructive and digital
Digital (162, 197, 320)-net over F2, using
- 22 times duplication [i] based on digital (160, 195, 320)-net over F2, using
- t-expansion [i] based on digital (159, 195, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 39, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 39, 64)-net over F32, using
- t-expansion [i] based on digital (159, 195, 320)-net over F2, using
(162, 162+35, 828)-Net over F2 — Digital
Digital (162, 197, 828)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2197, 828, F2, 2, 35) (dual of [(828, 2), 1459, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2197, 1039, F2, 2, 35) (dual of [(1039, 2), 1881, 36]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2196, 1039, F2, 2, 35) (dual of [(1039, 2), 1882, 36]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2194, 1038, F2, 2, 35) (dual of [(1038, 2), 1882, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2194, 2076, F2, 35) (dual of [2076, 1882, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(2194, 2076, F2, 35) (dual of [2076, 1882, 36]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2194, 1038, F2, 2, 35) (dual of [(1038, 2), 1882, 36]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2196, 1039, F2, 2, 35) (dual of [(1039, 2), 1882, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2197, 1039, F2, 2, 35) (dual of [(1039, 2), 1881, 36]-NRT-code), using
(162, 162+35, 21190)-Net in Base 2 — Upper bound on s
There is no (162, 197, 21191)-net in base 2, because
- 1 times m-reduction [i] would yield (162, 196, 21191)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 100509 345290 545946 872275 278107 459739 254537 621712 718345 905632 > 2196 [i]