Best Known (57−14, 57, s)-Nets in Base 2
(57−14, 57, 84)-Net over F2 — Constructive and digital
Digital (43, 57, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 19, 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
(57−14, 57, 132)-Net over F2 — Digital
Digital (43, 57, 132)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(257, 132, F2, 2, 14) (dual of [(132, 2), 207, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(257, 264, F2, 14) (dual of [264, 207, 15]-code), using
- 1 times truncation [i] based on linear OA(258, 265, F2, 15) (dual of [265, 207, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(257, 256, F2, 15) (dual of [256, 199, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(249, 256, F2, 13) (dual of [256, 207, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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(258, 265, F2, 15) (dual of [265, 207, 16]-code), using
- OOA 2-folding [i] based on linear OA(257, 264, F2, 14) (dual of [264, 207, 15]-code), using
(57−14, 57, 945)-Net in Base 2 — Upper bound on s
There is no (43, 57, 946)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 144753 497280 290504 > 257 [i]