Best Known (43, 43+12, s)-Nets in Base 128
(43, 43+12, 1398379)-Net over F128 — Constructive and digital
Digital (43, 55, 1398379)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (4, 10, 279)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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, 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 (0, 3, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- 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 (4, 10, 279)-net over F128, using
(43, 43+12, 2097151)-Net in Base 128 — Constructive
(43, 55, 2097151)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (10, 16, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12816, 699051, F128, 6, 6) (dual of [(699051, 6), 4194290, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12816, 2097153, F128, 6) (dual of [2097153, 2097137, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(12816, 2097153, F128, 6) (dual of [2097153, 2097137, 7]-code), using
- net defined by OOA [i] based on linear OOA(12816, 699051, F128, 6, 6) (dual of [(699051, 6), 4194290, 7]-NRT-code), 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
- digital (10, 16, 699051)-net over F128, using
(43, 43+12, large)-Net over F128 — Digital
Digital (43, 55, large)-net over F128, using
- 4 times m-reduction [i] based on digital (43, 59, large)-net over F128, using
(43, 43+12, large)-Net in Base 128 — Upper bound on s
There is no (43, 55, large)-net in base 128, because
- 10 times m-reduction [i] would yield (43, 45, large)-net in base 128, but