Best Known (109−19, 109, s)-Nets in Base 2
(109−19, 109, 455)-Net over F2 — Constructive and digital
Digital (90, 109, 455)-net over F2, using
- net defined by OOA [i] based on linear OOA(2109, 455, F2, 19, 19) (dual of [(455, 19), 8536, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using
(109−19, 109, 1024)-Net over F2 — Digital
Digital (90, 109, 1024)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2109, 1024, F2, 4, 19) (dual of [(1024, 4), 3987, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 4-folding [i] based on linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using
(109−19, 109, 16973)-Net in Base 2 — Upper bound on s
There is no (90, 109, 16974)-net in base 2, because
- 1 times m-reduction [i] would yield (90, 108, 16974)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 324 553373 877857 514655 198654 021857 > 2108 [i]