Best Known (56, 70, s)-Nets in Base 2
(56, 70, 165)-Net over F2 — Constructive and digital
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
(56, 70, 341)-Net over F2 — Digital
Digital (56, 70, 341)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(270, 341, F2, 2, 14) (dual of [(341, 2), 612, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(270, 511, F2, 2, 14) (dual of [(511, 2), 952, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(270, 1022, F2, 14) (dual of [1022, 952, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(270, 1023, F2, 14) (dual of [1023, 953, 15]-code), using
- 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]
- discarding factors / shortening the dual code based on linear OA(270, 1023, F2, 14) (dual of [1023, 953, 15]-code), using
- OOA 2-folding [i] based on linear OA(270, 1022, F2, 14) (dual of [1022, 952, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(270, 511, F2, 2, 14) (dual of [(511, 2), 952, 15]-NRT-code), using
(56, 70, 3451)-Net in Base 2 — Upper bound on s
There is no (56, 70, 3452)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1182 626927 139439 909146 > 270 [i]