Best Known (29, 29+30, s)-Nets in Base 64
(29, 29+30, 513)-Net over F64 — Constructive and digital
Digital (29, 59, 513)-net over F64, using
- t-expansion [i] based on digital (28, 59, 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
(29, 29+30, 1288)-Net over F64 — Digital
Digital (29, 59, 1288)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6459, 1288, F64, 3, 30) (dual of [(1288, 3), 3805, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6459, 1366, F64, 3, 30) (dual of [(1366, 3), 4039, 31]-NRT-code), using
- OOA 3-folding [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
- OOA 3-folding [i] based on linear OA(6459, 4098, F64, 30) (dual of [4098, 4039, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(6459, 1366, F64, 3, 30) (dual of [(1366, 3), 4039, 31]-NRT-code), using
(29, 29+30, 1296377)-Net in Base 64 — Upper bound on s
There is no (29, 59, 1296378)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 36696 089552 006804 378231 138060 208233 707680 031137 642145 803595 984511 599800 781016 673490 221784 533576 468610 532344 > 6459 [i]