Best Known (41−21, 41, s)-Nets in Base 128
(41−21, 41, 1638)-Net over F128 — Constructive and digital
Digital (20, 41, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
(41−21, 41, 3989)-Net over F128 — Digital
Digital (20, 41, 3989)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12841, 3989, F128, 4, 21) (dual of [(3989, 4), 15915, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12841, 4096, F128, 4, 21) (dual of [(4096, 4), 16343, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 4-folding [i] based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(12841, 4096, F128, 4, 21) (dual of [(4096, 4), 16343, 22]-NRT-code), using
(41−21, 41, large)-Net in Base 128 — Upper bound on s
There is no (20, 41, large)-net in base 128, because
- 19 times m-reduction [i] would yield (20, 22, large)-net in base 128, but