Best Known (51, 70, s)-Nets in Base 128
(51, 70, 233317)-Net over F128 — Constructive and digital
Digital (51, 70, 233317)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 300)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 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 (1, 10, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128 (see above)
- digital (1, 5, 150)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (36, 55, 233017)-net over F128, using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- digital (6, 15, 300)-net over F128, using
(51, 70, 932067)-Net in Base 128 — Constructive
(51, 70, 932067)-net in base 128, using
- 1284 times duplication [i] based on (47, 66, 932067)-net in base 128, using
- net defined by OOA [i] based on OOA(12866, 932067, S128, 21, 19), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12866, 2796202, S128, 3, 19), using
- 1 times NRT-code embedding in larger space [i] based on OOA(12863, 2796201, S128, 3, 19), using
- OOA 3-folding [i] 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
- OOA 3-folding [i] based on OA(12863, large, S128, 19), using
- 1 times NRT-code embedding in larger space [i] based on OOA(12863, 2796201, S128, 3, 19), using
- OOA 3-folding and stacking with additional row [i] based on OOA(12866, 2796202, S128, 3, 19), using
- net defined by OOA [i] based on OOA(12866, 932067, S128, 21, 19), using
(51, 70, large)-Net over F128 — Digital
Digital (51, 70, large)-net over F128, using
(51, 70, large)-Net in Base 128 — Upper bound on s
There is no (51, 70, large)-net in base 128, because
- 17 times m-reduction [i] would yield (51, 53, large)-net in base 128, but