Best Known (39, 39+23, s)-Nets in Base 128
(39, 39+23, 1768)-Net over F128 — Constructive and digital
Digital (39, 62, 1768)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 279)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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, 12, 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, 5, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (22, 45, 1489)-net over F128, using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12845, 16380, F128, 23) (dual of [16380, 16335, 24]-code), using
- net defined by OOA [i] based on linear OOA(12845, 1489, F128, 23, 23) (dual of [(1489, 23), 34202, 24]-NRT-code), using
- digital (6, 17, 279)-net over F128, using
(39, 39+23, 5960)-Net in Base 128 — Constructive
(39, 62, 5960)-net in base 128, using
- 1281 times duplication [i] based on (38, 61, 5960)-net in base 128, using
- net defined by OOA [i] based on OOA(12861, 5960, S128, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12861, 65561, S128, 23), using
- discarding factors based on OA(12861, 65562, S128, 23), using
- discarding parts of the base [i] based on linear OA(25653, 65562, F256, 23) (dual of [65562, 65509, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2568, 26, F256, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,256)), using
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- Reed–Solomon code RS(248,256) [i]
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- discarding parts of the base [i] based on linear OA(25653, 65562, F256, 23) (dual of [65562, 65509, 24]-code), using
- discarding factors based on OA(12861, 65562, S128, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12861, 65561, S128, 23), using
- net defined by OOA [i] based on OOA(12861, 5960, S128, 23, 23), using
(39, 39+23, 61891)-Net over F128 — Digital
Digital (39, 62, 61891)-net over F128, using
(39, 39+23, large)-Net in Base 128 — Upper bound on s
There is no (39, 62, large)-net in base 128, because
- 21 times m-reduction [i] would yield (39, 41, large)-net in base 128, but