Best Known (58, 58+19, s)-Nets in Base 128
(58, 58+19, 932067)-Net over F128 — Constructive and digital
Digital (58, 77, 932067)-net over F128, using
- 1281 times duplication [i] based on digital (57, 76, 932067)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 932067, F128, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(12876, 2796202, F128, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(12873, 2796201, F128, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12873, large, F128, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(12873, large, F128, 19) (dual of [large, large−73, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(12873, 2796201, F128, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(12876, 2796202, F128, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(12876, 932067, F128, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
(58, 58+19, 932345)-Net in Base 128 — Constructive
(58, 77, 932345)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 279)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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
- digital (1, 10, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (0, 4, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- (44, 63, 932066)-net in base 128, using
- net defined by OOA [i] based on OOA(12863, 932066, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12863, 8388595, S128, 19), using
- discarding factors based on OA(12863, large, S128, 19), using
- discarding parts of the base [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding parts of the base [i] based on linear OA(25655, large, F256, 19) (dual of [large, large−55, 20]-code), using
- discarding factors based on OA(12863, large, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12863, 8388595, S128, 19), using
- net defined by OOA [i] based on OOA(12863, 932066, S128, 19, 19), using
- digital (5, 14, 279)-net over F128, using
(58, 58+19, large)-Net over F128 — Digital
Digital (58, 77, large)-net over F128, using
- t-expansion [i] based on digital (56, 77, large)-net over F128, using
(58, 58+19, large)-Net in Base 128 — Upper bound on s
There is no (58, 77, large)-net in base 128, because
- 17 times m-reduction [i] would yield (58, 60, large)-net in base 128, but