Best Known (34, 34+19, s)-Nets in Base 128
(34, 34+19, 2207)-Net over F128 — Constructive and digital
Digital (34, 53, 2207)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 387)-net over F128, using
- generalized (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 (0, 4, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 9, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 3, 129)-net over F128, using
- generalized (u, u+v)-construction [i] based on
- digital (18, 37, 1820)-net over F128, using
- net defined by OOA [i] based on linear OOA(12837, 1820, F128, 19, 19) (dual of [(1820, 19), 34543, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12837, 16381, F128, 19) (dual of [16381, 16344, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(12837, 16384, F128, 19) (dual of [16384, 16347, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12837, 16381, F128, 19) (dual of [16381, 16344, 20]-code), using
- net defined by OOA [i] based on linear OOA(12837, 1820, F128, 19, 19) (dual of [(1820, 19), 34543, 20]-NRT-code), using
- digital (7, 16, 387)-net over F128, using
(34, 34+19, 7431)-Net in Base 128 — Constructive
(34, 53, 7431)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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
- (24, 43, 7281)-net in base 128, using
- net defined by OOA [i] based on OOA(12843, 7281, S128, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12843, 65530, S128, 19), using
- discarding factors based on OA(12843, 65538, S128, 19), using
- discarding parts of the base [i] based on linear OA(25637, 65538, F256, 19) (dual of [65538, 65501, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(25637, 65536, F256, 19) (dual of [65536, 65499, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(25637, 65538, F256, 19) (dual of [65538, 65501, 20]-code), using
- discarding factors based on OA(12843, 65538, S128, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(12843, 65530, S128, 19), using
- net defined by OOA [i] based on OOA(12843, 7281, S128, 19, 19), using
- digital (1, 10, 150)-net over F128, using
(34, 34+19, 95264)-Net over F128 — Digital
Digital (34, 53, 95264)-net over F128, using
(34, 34+19, large)-Net in Base 128 — Upper bound on s
There is no (34, 53, large)-net in base 128, because
- 17 times m-reduction [i] would yield (34, 36, large)-net in base 128, but