Best Known (40−16, 40, s)-Nets in Base 64
(40−16, 40, 592)-Net over F64 — Constructive and digital
Digital (24, 40, 592)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (15, 31, 512)-net over F64, using
- net defined by OOA [i] based on linear OOA(6431, 512, F64, 16, 16) (dual of [(512, 16), 8161, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using
- net defined by OOA [i] based on linear OOA(6431, 512, F64, 16, 16) (dual of [(512, 16), 8161, 17]-NRT-code), using
- digital (1, 9, 80)-net over F64, using
(40−16, 40, 2049)-Net in Base 64 — Constructive
(24, 40, 2049)-net in base 64, using
- 641 times duplication [i] based on (23, 39, 2049)-net in base 64, using
- net defined by OOA [i] based on OOA(6439, 2049, S64, 16, 16), using
- OA 8-folding and stacking [i] based on OA(6439, 16392, S64, 16), using
- discarding parts of the base [i] based on linear OA(12833, 16392, F128, 16) (dual of [16392, 16359, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding parts of the base [i] based on linear OA(12833, 16392, F128, 16) (dual of [16392, 16359, 17]-code), using
- OA 8-folding and stacking [i] based on OA(6439, 16392, S64, 16), using
- net defined by OOA [i] based on OOA(6439, 2049, S64, 16, 16), using
(40−16, 40, 6689)-Net over F64 — Digital
Digital (24, 40, 6689)-net over F64, using
(40−16, 40, large)-Net in Base 64 — Upper bound on s
There is no (24, 40, large)-net in base 64, because
- 14 times m-reduction [i] would yield (24, 26, large)-net in base 64, but