Best Known (57−15, 57, s)-Nets in Base 2
(57−15, 57, 80)-Net over F2 — Constructive and digital
Digital (42, 57, 80)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (11, 18, 41)-net over F2, using
- digital (24, 39, 40)-net over F2, using
- 1 times m-reduction [i] based on digital (24, 40, 40)-net over F2, using
(57−15, 57, 111)-Net over F2 — Digital
Digital (42, 57, 111)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(257, 111, F2, 2, 15) (dual of [(111, 2), 165, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(257, 128, F2, 2, 15) (dual of [(128, 2), 199, 16]-NRT-code), using
- OOA 2-folding [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]
- OOA 2-folding [i] based on linear OA(257, 256, F2, 15) (dual of [256, 199, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(257, 128, F2, 2, 15) (dual of [(128, 2), 199, 16]-NRT-code), using
(57−15, 57, 855)-Net in Base 2 — Upper bound on s
There is no (42, 57, 856)-net in base 2, because
- 1 times m-reduction [i] would yield (42, 56, 856)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 72446 482858 511627 > 256 [i]