Best Known (39−16, 39, s)-Nets in Base 64
(39−16, 39, 577)-Net over F64 — Constructive and digital
Digital (23, 39, 577)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-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 (0, 8, 65)-net over F64, using
(39−16, 39, 2049)-Net in Base 64 — Constructive
(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
(39−16, 39, 5266)-Net over F64 — Digital
Digital (23, 39, 5266)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6439, 5266, F64, 16) (dual of [5266, 5227, 17]-code), using
- 1160 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 20 times 0, 1, 81 times 0, 1, 284 times 0, 1, 766 times 0) [i] based on linear OA(6431, 4098, F64, 16) (dual of [4098, 4067, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [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]
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- 1160 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 20 times 0, 1, 81 times 0, 1, 284 times 0, 1, 766 times 0) [i] based on linear OA(6431, 4098, F64, 16) (dual of [4098, 4067, 17]-code), using
(39−16, 39, large)-Net in Base 64 — Upper bound on s
There is no (23, 39, large)-net in base 64, because
- 14 times m-reduction [i] would yield (23, 25, large)-net in base 64, but