Best Known (165, 188, s)-Nets in Base 2
(165, 188, 11915)-Net over F2 — Constructive and digital
Digital (165, 188, 11915)-net over F2, using
- net defined by OOA [i] based on linear OOA(2188, 11915, F2, 23, 23) (dual of [(11915, 23), 273857, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2188, 131066, F2, 23) (dual of [131066, 130878, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2188, 131066, F2, 23) (dual of [131066, 130878, 24]-code), using
(165, 188, 18724)-Net over F2 — Digital
Digital (165, 188, 18724)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2188, 18724, F2, 7, 23) (dual of [(18724, 7), 130880, 24]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2188, 131068, F2, 23) (dual of [131068, 130880, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using
- OOA 7-folding [i] based on linear OA(2188, 131068, F2, 23) (dual of [131068, 130880, 24]-code), using
(165, 188, 643447)-Net in Base 2 — Upper bound on s
There is no (165, 188, 643448)-net in base 2, because
- 1 times m-reduction [i] would yield (165, 187, 643448)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 196 160282 670009 129575 861677 217926 497677 904575 422619 278734 > 2187 [i]