Best Known (48, 48+30, s)-Nets in Base 128
(48, 48+30, 1284)-Net over F128 — Constructive and digital
Digital (48, 78, 1284)-net over F128, using
- 1281 times duplication [i] based on digital (47, 77, 1284)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 18, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (29, 59, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12859, 1092, F128, 30, 30) (dual of [(1092, 30), 32701, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(12859, 16380, F128, 30) (dual of [16380, 16321, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(12859, 16380, F128, 30) (dual of [16380, 16321, 31]-code), using
- net defined by OOA [i] based on linear OOA(12859, 1092, F128, 30, 30) (dual of [(1092, 30), 32701, 31]-NRT-code), using
- digital (3, 18, 192)-net over F128, using
- (u, u+v)-construction [i] based on
(48, 48+30, 4371)-Net in Base 128 — Constructive
(48, 78, 4371)-net in base 128, using
- net defined by OOA [i] based on OOA(12878, 4371, S128, 30, 30), using
- OA 15-folding and stacking [i] based on OA(12878, 65565, S128, 30), using
- discarding parts of the base [i] based on linear OA(25668, 65565, F256, 30) (dual of [65565, 65497, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(19) [i] based on
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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 Ce(29) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(25668, 65565, F256, 30) (dual of [65565, 65497, 31]-code), using
- OA 15-folding and stacking [i] based on OA(12878, 65565, S128, 30), using
(48, 48+30, 42769)-Net over F128 — Digital
Digital (48, 78, 42769)-net over F128, using
(48, 48+30, large)-Net in Base 128 — Upper bound on s
There is no (48, 78, large)-net in base 128, because
- 28 times m-reduction [i] would yield (48, 50, large)-net in base 128, but