Best Known (49, 78, s)-Nets in Base 128
(49, 78, 1428)-Net over F128 — Constructive and digital
Digital (49, 78, 1428)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (0, 14, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 7, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (28, 57, 1170)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1170, F128, 29, 29) (dual of [(1170, 29), 33873, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12857, 16381, F128, 29) (dual of [16381, 16324, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12857, 16381, F128, 29) (dual of [16381, 16324, 30]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1170, F128, 29, 29) (dual of [(1170, 29), 33873, 30]-NRT-code), using
- digital (7, 21, 258)-net over F128, using
(49, 78, 4683)-Net in Base 128 — Constructive
(49, 78, 4683)-net in base 128, using
- 1282 times duplication [i] based on (47, 76, 4683)-net in base 128, using
- net defined by OOA [i] based on OOA(12876, 4683, S128, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12876, 65563, S128, 29), using
- discarding factors based on OA(12876, 65566, S128, 29), using
- discarding parts of the base [i] based on linear OA(25666, 65566, F256, 29) (dual of [65566, 65500, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2569, 29, F256, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,256)), using
- discarding factors / shortening the dual code based on linear OA(2569, 256, F256, 9) (dual of [256, 247, 10]-code or 256-arc in PG(8,256)), using
- Reed–Solomon code RS(247,256) [i]
- discarding factors / shortening the dual code based on linear OA(2569, 256, F256, 9) (dual of [256, 247, 10]-code or 256-arc in PG(8,256)), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- discarding parts of the base [i] based on linear OA(25666, 65566, F256, 29) (dual of [65566, 65500, 30]-code), using
- discarding factors based on OA(12876, 65566, S128, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12876, 65563, S128, 29), using
- net defined by OOA [i] based on OOA(12876, 4683, S128, 29, 29), using
(49, 78, 65975)-Net over F128 — Digital
Digital (49, 78, 65975)-net over F128, using
(49, 78, large)-Net in Base 128 — Upper bound on s
There is no (49, 78, large)-net in base 128, because
- 27 times m-reduction [i] would yield (49, 51, large)-net in base 128, but