Best Known (38, 38+21, s)-Nets in Base 128
(38, 38+21, 2025)-Net over F128 — Constructive and digital
Digital (38, 59, 2025)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (8, 18, 387)-net over F128, using
- 1 times m-reduction [i] based on digital (8, 19, 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, 5, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 11, 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
- 1 times m-reduction [i] based on digital (8, 19, 387)-net over F128, using
- digital (20, 41, 1638)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12841, 16381, F128, 21) (dual of [16381, 16340, 22]-code), using
- net defined by OOA [i] based on linear OOA(12841, 1638, F128, 21, 21) (dual of [(1638, 21), 34357, 22]-NRT-code), using
- digital (8, 18, 387)-net over F128, using
(38, 38+21, 6810)-Net in Base 128 — Constructive
(38, 59, 6810)-net in base 128, using
- (u, u+v)-construction [i] based on
- (2, 12, 257)-net in base 128, using
- 4 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- base change [i] based on digital (0, 14, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 14, 257)-net over F256, using
- 4 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
- (26, 47, 6553)-net in base 128, using
- net defined by OOA [i] based on OOA(12847, 6553, S128, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12847, 65531, S128, 21), using
- discarding factors based on OA(12847, 65538, S128, 21), using
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(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(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(25641, 65538, F256, 21) (dual of [65538, 65497, 22]-code), using
- discarding factors based on OA(12847, 65538, S128, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(12847, 65531, S128, 21), using
- net defined by OOA [i] based on OOA(12847, 6553, S128, 21, 21), using
- (2, 12, 257)-net in base 128, using
(38, 38+21, 107600)-Net over F128 — Digital
Digital (38, 59, 107600)-net over F128, using
(38, 38+21, large)-Net in Base 128 — Upper bound on s
There is no (38, 59, large)-net in base 128, because
- 19 times m-reduction [i] would yield (38, 40, large)-net in base 128, but