Best Known (3, 3+4, s)-Nets in Base 64
(3, 3+4, 2049)-Net over F64 — Constructive and digital
Digital (3, 7, 2049)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
(3, 3+4, 4098)-Net over F64 — Digital
Digital (3, 7, 4098)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 4098, F64, 4, 4) (dual of [(4098, 4), 16385, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(647, 4098, F64, 3, 4) (dual of [(4098, 3), 12287, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- appending kth column [i] based on linear OOA(647, 4098, F64, 3, 4) (dual of [(4098, 3), 12287, 5]-NRT-code), using
(3, 3+4, 8128)-Net in Base 64 — Constructive
(3, 7, 8128)-net in base 64, using
- base change [i] based on digital (2, 6, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- 1 times truncation [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
(3, 3+4, 47075)-Net in Base 64 — Upper bound on s
There is no (3, 7, 47076)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 4 398048 584371 > 647 [i]