Best Known (37−19, 37, s)-Nets in Base 128
(37−19, 37, 1820)-Net over F128 — Constructive and digital
Digital (18, 37, 1820)-net over F128, using
- net defined by OOA [i] based on linear OOA(12837, 1820, F128, 19, 19) (dual of [(1820, 19), 34543, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12837, 16381, F128, 19) (dual of [16381, 16344, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12837, 16381, F128, 19) (dual of [16381, 16344, 20]-code), using
(37−19, 37, 4096)-Net over F128 — Digital
Digital (18, 37, 4096)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12837, 4096, F128, 4, 19) (dual of [(4096, 4), 16347, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 4-folding [i] based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
(37−19, 37, large)-Net in Base 128 — Upper bound on s
There is no (18, 37, large)-net in base 128, because
- 17 times m-reduction [i] would yield (18, 20, large)-net in base 128, but