Best Known (217−36, 217, s)-Nets in Base 2
(217−36, 217, 490)-Net over F2 — Constructive and digital
Digital (181, 217, 490)-net over F2, using
- 22 times duplication [i] based on digital (179, 215, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 43, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 43, 98)-net over F32, using
(217−36, 217, 1272)-Net over F2 — Digital
Digital (181, 217, 1272)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 1272, F2, 3, 36) (dual of [(1272, 3), 3599, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2217, 1369, F2, 3, 36) (dual of [(1369, 3), 3890, 37]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2217, 4107, F2, 36) (dual of [4107, 3890, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 4108, F2, 36) (dual of [4108, 3891, 37]-code), using
- 1 times truncation [i] based on linear OA(2218, 4109, F2, 37) (dual of [4109, 3891, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] 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]
- linear OA(2205, 4096, F2, 35) (dual of [4096, 3891, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 1 times truncation [i] based on linear OA(2218, 4109, F2, 37) (dual of [4109, 3891, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 4108, F2, 36) (dual of [4108, 3891, 37]-code), using
- OOA 3-folding [i] based on linear OA(2217, 4107, F2, 36) (dual of [4107, 3890, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(2217, 1369, F2, 3, 36) (dual of [(1369, 3), 3890, 37]-NRT-code), using
(217−36, 217, 32125)-Net in Base 2 — Upper bound on s
There is no (181, 217, 32126)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 210630 864107 067034 434533 195070 217533 692864 039536 810045 339167 196294 > 2217 [i]