Best Known (69−30, 69, s)-Nets in Base 64
(69−30, 69, 513)-Net over F64 — Constructive and digital
Digital (39, 69, 513)-net over F64, using
- t-expansion [i] based on digital (28, 69, 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
(69−30, 69, 1092)-Net in Base 64 — Constructive
(39, 69, 1092)-net in base 64, using
- net defined by OOA [i] based on OOA(6469, 1092, S64, 30, 30), using
- OA 15-folding and stacking [i] based on OA(6469, 16380, S64, 30), using
- discarding factors based on OA(6469, 16386, S64, 30), using
- discarding parts of the base [i] based on linear OA(12859, 16386, F128, 30) (dual of [16386, 16327, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- discarding parts of the base [i] based on linear OA(12859, 16386, F128, 30) (dual of [16386, 16327, 31]-code), using
- discarding factors based on OA(6469, 16386, S64, 30), using
- OA 15-folding and stacking [i] based on OA(6469, 16380, S64, 30), using
(69−30, 69, 4305)-Net over F64 — Digital
Digital (39, 69, 4305)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6469, 4305, F64, 30) (dual of [4305, 4236, 31]-code), using
- 197 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 49 times 0, 1, 111 times 0) [i] based on linear OA(6459, 4098, F64, 30) (dual of [4098, 4039, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(6457, 4096, F64, 29) (dual of [4096, 4039, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(29) ⊂ Ce(28) [i] based on
- 197 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 49 times 0, 1, 111 times 0) [i] based on linear OA(6459, 4098, F64, 30) (dual of [4098, 4039, 31]-code), using
(69−30, 69, large)-Net in Base 64 — Upper bound on s
There is no (39, 69, large)-net in base 64, because
- 28 times m-reduction [i] would yield (39, 41, large)-net in base 64, but