Best Known (57, 57+20, s)-Nets in Base 128
(57, 57+20, 838860)-Net over F128 — Constructive and digital
Digital (57, 77, 838860)-net over F128, using
- net defined by OOA [i] based on linear OOA(12877, 838860, F128, 20, 20) (dual of [(838860, 20), 16777123, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12877, 8388600, F128, 20) (dual of [8388600, 8388523, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12877, large, F128, 20) (dual of [large, large−77, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12877, large, F128, 20) (dual of [large, large−77, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12877, 8388600, F128, 20) (dual of [8388600, 8388523, 21]-code), using
(57, 57+20, 838989)-Net in Base 128 — Constructive
(57, 77, 838989)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- (47, 67, 838860)-net in base 128, using
- net defined by OOA [i] based on OOA(12867, 838860, S128, 20, 20), using
- OA 10-folding and stacking [i] based on OA(12867, 8388600, S128, 20), using
- discarding factors based on OA(12867, large, S128, 20), using
- discarding parts of the base [i] based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding parts of the base [i] based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- discarding factors based on OA(12867, large, S128, 20), using
- OA 10-folding and stacking [i] based on OA(12867, 8388600, S128, 20), using
- net defined by OOA [i] based on OOA(12867, 838860, S128, 20, 20), using
- digital (0, 10, 129)-net over F128, using
(57, 57+20, large)-Net over F128 — Digital
Digital (57, 77, large)-net over F128, using
- t-expansion [i] based on digital (56, 77, large)-net over F128, using
(57, 57+20, large)-Net in Base 128 — Upper bound on s
There is no (57, 77, large)-net in base 128, because
- 18 times m-reduction [i] would yield (57, 59, large)-net in base 128, but