Best Known (19−8, 19, s)-Nets in Base 64
(19−8, 19, 1089)-Net over F64 — Constructive and digital
Digital (11, 19, 1089)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (7, 15, 1024)-net over F64, using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- digital (0, 4, 65)-net over F64, using
(19−8, 19, 4097)-Net in Base 64 — Constructive
(11, 19, 4097)-net in base 64, using
- net defined by OOA [i] based on OOA(6419, 4097, S64, 8, 8), using
- OA 4-folding and stacking [i] based on OA(6419, 16388, S64, 8), using
- discarding factors based on OA(6419, 16389, S64, 8), using
- discarding parts of the base [i] based on linear OA(12816, 16389, F128, 8) (dual of [16389, 16373, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12816, 16389, F128, 8) (dual of [16389, 16373, 9]-code), using
- discarding factors based on OA(6419, 16389, S64, 8), using
- OA 4-folding and stacking [i] based on OA(6419, 16388, S64, 8), using
(19−8, 19, 4971)-Net over F64 — Digital
Digital (11, 19, 4971)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6419, 4971, F64, 8) (dual of [4971, 4952, 9]-code), using
- 869 step Varšamov–Edel lengthening with (ri) = (2, 12 times 0, 1, 110 times 0, 1, 744 times 0) [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- 869 step Varšamov–Edel lengthening with (ri) = (2, 12 times 0, 1, 110 times 0, 1, 744 times 0) [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
(19−8, 19, large)-Net in Base 64 — Upper bound on s
There is no (11, 19, large)-net in base 64, because
- 6 times m-reduction [i] would yield (11, 13, large)-net in base 64, but