Best Known (64−16, 64, s)-Nets in Base 2
(64−16, 64, 84)-Net over F2 — Constructive and digital
Digital (48, 64, 84)-net over F2, using
- 21 times duplication [i] based on digital (47, 63, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 21, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- trace code for nets [i] based on digital (5, 21, 28)-net over F8, using
(64−16, 64, 128)-Net over F2 — Digital
Digital (48, 64, 128)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(264, 128, F2, 2, 16) (dual of [(128, 2), 192, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(264, 256, F2, 16) (dual of [256, 192, 17]-code), using
- 1 times truncation [i] based on linear OA(265, 257, F2, 17) (dual of [257, 192, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 216−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(265, 257, F2, 17) (dual of [257, 192, 18]-code), using
- OOA 2-folding [i] based on linear OA(264, 256, F2, 16) (dual of [256, 192, 17]-code), using
(64−16, 64, 952)-Net in Base 2 — Upper bound on s
There is no (48, 64, 953)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 18 564345 698886 502091 > 264 [i]