Best Known (70−30, 70, s)-Nets in Base 64
(70−30, 70, 513)-Net over F64 — Constructive and digital
Digital (40, 70, 513)-net over F64, using
- t-expansion [i] based on digital (28, 70, 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
(70−30, 70, 1092)-Net in Base 64 — Constructive
(40, 70, 1092)-net in base 64, using
- base change [i] based on digital (30, 60, 1092)-net over F128, using
- 1 times m-reduction [i] based on digital (30, 61, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12861, 16381, F128, 31) (dual of [16381, 16320, 32]-code), using
- net defined by OOA [i] based on linear OOA(12861, 1092, F128, 31, 31) (dual of [(1092, 31), 33791, 32]-NRT-code), using
- 1 times m-reduction [i] based on digital (30, 61, 1092)-net over F128, using
(70−30, 70, 4544)-Net over F64 — Digital
Digital (40, 70, 4544)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6470, 4544, F64, 30) (dual of [4544, 4474, 31]-code), using
- 435 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, 1, 237 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
- 435 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, 1, 237 times 0) [i] based on linear OA(6459, 4098, F64, 30) (dual of [4098, 4039, 31]-code), using
(70−30, 70, large)-Net in Base 64 — Upper bound on s
There is no (40, 70, large)-net in base 64, because
- 28 times m-reduction [i] would yield (40, 42, large)-net in base 64, but