Best Known (106−15, 106, s)-Nets in Base 2
(106−15, 106, 4681)-Net over F2 — Constructive and digital
Digital (91, 106, 4681)-net over F2, using
- net defined by OOA [i] based on linear OOA(2106, 4681, F2, 15, 15) (dual of [(4681, 15), 70109, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using
(106−15, 106, 6553)-Net over F2 — Digital
Digital (91, 106, 6553)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2106, 6553, F2, 5, 15) (dual of [(6553, 5), 32659, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2106, 32765, F2, 15) (dual of [32765, 32659, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2106, 32768, F2, 15) (dual of [32768, 32662, 16]-code), using
- OOA 5-folding [i] based on linear OA(2106, 32765, F2, 15) (dual of [32765, 32659, 16]-code), using
(106−15, 106, 110746)-Net in Base 2 — Upper bound on s
There is no (91, 106, 110747)-net in base 2, because
- 1 times m-reduction [i] would yield (91, 105, 110747)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 40 566518 678698 923108 552956 139708 > 2105 [i]