Best Known (41, 41+12, s)-Nets in Base 128
(41, 41+12, 1398250)-Net over F128 — Constructive and digital
Digital (41, 53, 1398250)-net over F128, using
- 1281 times duplication [i] based on digital (40, 52, 1398250)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (33, 45, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1398100, F128, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12845, 8388600, F128, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(12845, large, F128, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(12845, 8388600, F128, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1398100, F128, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (1, 7, 150)-net over F128, using
- (u, u+v)-construction [i] based on
(41, 41+12, 1419947)-Net in Base 128 — Constructive
(41, 53, 1419947)-net in base 128, using
- (u, u+v)-construction [i] based on
- (8, 14, 21847)-net in base 128, using
- net defined by OOA [i] based on OOA(12814, 21847, S128, 6, 6), using
- OA 3-folding and stacking [i] based on OA(12814, 65541, S128, 6), using
- discarding parts of the base [i] based on linear OA(25612, 65541, F256, 6) (dual of [65541, 65529, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(25612, 65541, F256, 6) (dual of [65541, 65529, 7]-code), using
- OA 3-folding and stacking [i] based on OA(12814, 65541, S128, 6), using
- net defined by OOA [i] based on OOA(12814, 21847, S128, 6, 6), using
- (27, 39, 1398100)-net in base 128, using
- net defined by OOA [i] based on OOA(12839, 1398100, S128, 12, 12), using
- OA 6-folding and stacking [i] based on OA(12839, 8388600, S128, 12), using
- discarding factors based on OA(12839, large, S128, 12), using
- discarding parts of the base [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding parts of the base [i] based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- discarding factors based on OA(12839, large, S128, 12), using
- OA 6-folding and stacking [i] based on OA(12839, 8388600, S128, 12), using
- net defined by OOA [i] based on OOA(12839, 1398100, S128, 12, 12), using
- (8, 14, 21847)-net in base 128, using
(41, 41+12, large)-Net over F128 — Digital
Digital (41, 53, large)-net over F128, using
- t-expansion [i] based on digital (40, 53, large)-net over F128, using
- 2 times m-reduction [i] based on digital (40, 55, large)-net over F128, using
(41, 41+12, large)-Net in Base 128 — Upper bound on s
There is no (41, 53, large)-net in base 128, because
- 10 times m-reduction [i] would yield (41, 43, large)-net in base 128, but