Best Known (50−17, 50, s)-Nets in Base 128
(50−17, 50, 262144)-Net over F128 — Constructive and digital
Digital (33, 50, 262144)-net over F128, using
- 1281 times duplication [i] based on digital (32, 49, 262144)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 262144, F128, 17, 17) (dual of [(262144, 17), 4456399, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using
- net defined by OOA [i] based on linear OOA(12849, 262144, F128, 17, 17) (dual of [(262144, 17), 4456399, 18]-NRT-code), using
(50−17, 50, 798676)-Net over F128 — Digital
Digital (33, 50, 798676)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12850, 798676, F128, 2, 17) (dual of [(798676, 2), 1597302, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12850, 1048580, F128, 2, 17) (dual of [(1048580, 2), 2097110, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- OOA 2-folding [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(12850, 1048580, F128, 2, 17) (dual of [(1048580, 2), 2097110, 18]-NRT-code), using
(50−17, 50, large)-Net in Base 128 — Upper bound on s
There is no (33, 50, large)-net in base 128, because
- 15 times m-reduction [i] would yield (33, 35, large)-net in base 128, but