Best Known (71−14, 71, s)-Nets in Base 2
(71−14, 71, 165)-Net over F2 — Constructive and digital
Digital (57, 71, 165)-net over F2, using
- 21 times duplication [i] based on digital (56, 70, 165)-net over F2, using
- trace code for nets [i] based on digital (0, 14, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- trace code for nets [i] based on digital (0, 14, 33)-net over F32, using
(71−14, 71, 364)-Net over F2 — Digital
Digital (57, 71, 364)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 364, F2, 2, 14) (dual of [(364, 2), 657, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(271, 517, F2, 2, 14) (dual of [(517, 2), 963, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(271, 1034, F2, 14) (dual of [1034, 963, 15]-code), using
- 1 times truncation [i] based on linear OA(272, 1035, F2, 15) (dual of [1035, 963, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(271, 1024, F2, 15) (dual of [1024, 953, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(261, 1024, F2, 13) (dual of [1024, 963, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(272, 1035, F2, 15) (dual of [1035, 963, 16]-code), using
- OOA 2-folding [i] based on linear OA(271, 1034, F2, 14) (dual of [1034, 963, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(271, 517, F2, 2, 14) (dual of [(517, 2), 963, 15]-NRT-code), using
(71−14, 71, 3811)-Net in Base 2 — Upper bound on s
There is no (57, 71, 3812)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2363 714922 354410 711352 > 271 [i]