Best Known (28, 28+28, s)-Nets in Base 64
(28, 28+28, 513)-Net over F64 — Constructive and digital
Digital (28, 56, 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
(28, 28+28, 514)-Net in Base 64 — Constructive
(28, 56, 514)-net in base 64, using
- base change [i] based on digital (14, 42, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 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, 28, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 14, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(28, 28+28, 1367)-Net over F64 — Digital
Digital (28, 56, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6456, 1367, F64, 3, 28) (dual of [(1367, 3), 4045, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6456, 4101, F64, 28) (dual of [4101, 4045, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(6455, 4096, F64, 28) (dual of [4096, 4041, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(27) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(6456, 4101, F64, 28) (dual of [4101, 4045, 29]-code), using
(28, 28+28, 1610034)-Net in Base 64 — Upper bound on s
There is no (28, 56, 1610035)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 139984 247015 582005 587165 353772 979803 410818 196265 304998 772795 548026 966047 821171 229825 187971 844349 128620 > 6456 [i]