Best Known (40−15, 40, s)-Nets in Base 128
(40−15, 40, 2619)-Net over F128 — Constructive and digital
Digital (25, 40, 2619)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 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, 8, 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 (14, 29, 2340)-net over F128, using
- net defined by OOA [i] based on linear OOA(12829, 2340, F128, 15, 15) (dual of [(2340, 15), 35071, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12829, 16381, F128, 15) (dual of [16381, 16352, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(12829, 16384, F128, 15) (dual of [16384, 16355, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12829, 16381, F128, 15) (dual of [16381, 16352, 16]-code), using
- net defined by OOA [i] based on linear OOA(12829, 2340, F128, 15, 15) (dual of [(2340, 15), 35071, 16]-NRT-code), using
- digital (4, 11, 279)-net over F128, using
(40−15, 40, 9365)-Net in Base 128 — Constructive
(25, 40, 9365)-net in base 128, using
- base change [i] based on digital (20, 35, 9365)-net over F256, using
- net defined by OOA [i] based on linear OOA(25635, 9365, F256, 15, 15) (dual of [(9365, 15), 140440, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25635, 65556, F256, 15) (dual of [65556, 65521, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(7) [i] based on
- linear OA(25629, 65536, F256, 15) (dual of [65536, 65507, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2566, 20, F256, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,256)), using
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- Reed–Solomon code RS(250,256) [i]
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- construction X applied to Ce(14) ⊂ Ce(7) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(25635, 65556, F256, 15) (dual of [65556, 65521, 16]-code), using
- net defined by OOA [i] based on linear OOA(25635, 9365, F256, 15, 15) (dual of [(9365, 15), 140440, 16]-NRT-code), using
(40−15, 40, 49925)-Net over F128 — Digital
Digital (25, 40, 49925)-net over F128, using
(40−15, 40, large)-Net in Base 128 — Upper bound on s
There is no (25, 40, large)-net in base 128, because
- 13 times m-reduction [i] would yield (25, 27, large)-net in base 128, but