Best Known (84−29, 84, s)-Nets in Base 64
(84−29, 84, 1026)-Net over F64 — Constructive and digital
Digital (55, 84, 1026)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (13, 27, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- digital (28, 57, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- digital (13, 27, 585)-net over F64, using
(84−29, 84, 4682)-Net in Base 64 — Constructive
(55, 84, 4682)-net in base 64, using
- base change [i] based on digital (34, 63, 4682)-net over F256, using
- 2562 times duplication [i] based on digital (32, 61, 4682)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 4682, F256, 29, 29) (dual of [(4682, 29), 135717, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25661, 65549, F256, 29) (dual of [65549, 65488, 30]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(25661, 65550, F256, 29) (dual of [65550, 65489, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25661, 65549, F256, 29) (dual of [65549, 65488, 30]-code), using
- net defined by OOA [i] based on linear OOA(25661, 4682, F256, 29, 29) (dual of [(4682, 29), 135717, 30]-NRT-code), using
- 2562 times duplication [i] based on digital (32, 61, 4682)-net over F256, using
(84−29, 84, 47026)-Net over F64 — Digital
Digital (55, 84, 47026)-net over F64, using
(84−29, 84, large)-Net in Base 64 — Upper bound on s
There is no (55, 84, large)-net in base 64, because
- 27 times m-reduction [i] would yield (55, 57, large)-net in base 64, but