Best Known (57, 57+19, s)-Nets in Base 2
(57, 57+19, 84)-Net over F2 — Constructive and digital
Digital (57, 76, 84)-net over F2, using
- 2 times m-reduction [i] based on digital (57, 78, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 26, 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, 26, 28)-net over F8, using
(57, 57+19, 137)-Net over F2 — Digital
Digital (57, 76, 137)-net over F2, using
- 21 times duplication [i] based on digital (56, 75, 137)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(275, 137, F2, 2, 19) (dual of [(137, 2), 199, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(275, 274, F2, 19) (dual of [274, 199, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(269, 256, F2, 19) (dual of [256, 187, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(26, 18, F2, 3) (dual of [18, 12, 4]-code or 18-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(275, 274, F2, 19) (dual of [274, 199, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(275, 137, F2, 2, 19) (dual of [(137, 2), 199, 20]-NRT-code), using
(57, 57+19, 1324)-Net in Base 2 — Upper bound on s
There is no (57, 76, 1325)-net in base 2, because
- 1 times m-reduction [i] would yield (57, 75, 1325)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 37859 800219 956064 080866 > 275 [i]