Best Known (88−44, 88, s)-Nets in Base 64
(88−44, 88, 513)-Net over F64 — Constructive and digital
Digital (44, 88, 513)-net over F64, using
- t-expansion [i] based on digital (28, 88, 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
(88−44, 88, 514)-Net in Base 64 — Constructive
(44, 88, 514)-net in base 64, using
- base change [i] based on digital (22, 66, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 22, 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, 44, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 22, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(88−44, 88, 1554)-Net over F64 — Digital
Digital (44, 88, 1554)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6488, 1554, F64, 2, 44) (dual of [(1554, 2), 3020, 45]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6488, 2050, F64, 2, 44) (dual of [(2050, 2), 4012, 45]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6488, 4100, F64, 44) (dual of [4100, 4012, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(6488, 4101, F64, 44) (dual of [4101, 4013, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(41) [i] based on
- linear OA(6487, 4096, F64, 44) (dual of [4096, 4009, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(6483, 4096, F64, 42) (dual of [4096, 4013, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [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(43) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(6488, 4101, F64, 44) (dual of [4101, 4013, 45]-code), using
- OOA 2-folding [i] based on linear OA(6488, 4100, F64, 44) (dual of [4100, 4012, 45]-code), using
- discarding factors / shortening the dual code based on linear OOA(6488, 2050, F64, 2, 44) (dual of [(2050, 2), 4012, 45]-NRT-code), using
(88−44, 88, 2411184)-Net in Base 64 — Upper bound on s
There is no (44, 88, 2411185)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 878 699171 978514 958738 015003 479539 512914 598524 694539 171677 037430 132119 052181 131186 074336 829971 864220 440442 516296 962909 426125 874232 474477 494796 935046 375394 086064 > 6488 [i]