Best Known (69−28, 69, s)-Nets in Base 128
(69−28, 69, 1299)-Net over F128 — Constructive and digital
Digital (41, 69, 1299)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 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 (27, 55, 1170)-net over F128, using
- net defined by OOA [i] based on linear OOA(12855, 1170, F128, 28, 28) (dual of [(1170, 28), 32705, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(12855, 16380, F128, 28) (dual of [16380, 16325, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(12855, 16380, F128, 28) (dual of [16380, 16325, 29]-code), using
- net defined by OOA [i] based on linear OOA(12855, 1170, F128, 28, 28) (dual of [(1170, 28), 32705, 29]-NRT-code), using
- digital (0, 14, 129)-net over F128, using
(69−28, 69, 4682)-Net in Base 128 — Constructive
(41, 69, 4682)-net in base 128, using
- 1 times m-reduction [i] based on (41, 70, 4682)-net in base 128, using
- net defined by OOA [i] based on OOA(12870, 4682, S128, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12870, 65549, S128, 29), using
- discarding factors based on OA(12870, 65550, S128, 29), using
- discarding parts of the base [i] based on linear OA(25661, 65550, F256, 29) (dual of [65550, 65489, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(25647, 65536, F256, 24) (dual of [65536, 65489, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(25661, 65550, F256, 29) (dual of [65550, 65489, 30]-code), using
- discarding factors based on OA(12870, 65550, S128, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(12870, 65549, S128, 29), using
- net defined by OOA [i] based on OOA(12870, 4682, S128, 29, 29), using
(69−28, 69, 20892)-Net over F128 — Digital
Digital (41, 69, 20892)-net over F128, using
(69−28, 69, large)-Net in Base 128 — Upper bound on s
There is no (41, 69, large)-net in base 128, because
- 26 times m-reduction [i] would yield (41, 43, large)-net in base 128, but