Best Known (32−16, 32, s)-Nets in Base 64
(32−16, 32, 512)-Net over F64 — Constructive and digital
Digital (16, 32, 512)-net over F64, using
- 1 times m-reduction [i] based on digital (16, 33, 512)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
(32−16, 32, 514)-Net in Base 64 — Constructive
(16, 32, 514)-net in base 64, using
- base change [i] based on digital (8, 24, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (0, 16, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 8, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(32−16, 32, 1367)-Net over F64 — Digital
Digital (16, 32, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6432, 1367, F64, 3, 16) (dual of [(1367, 3), 4069, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6432, 4101, F64, 16) (dual of [4101, 4069, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(6432, 4101, F64, 16) (dual of [4101, 4069, 17]-code), using
(32−16, 32, 1002461)-Net in Base 64 — Upper bound on s
There is no (16, 32, 1002462)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 6277 110793 704836 896664 457014 278421 166965 456229 606618 532585 > 6432 [i]