Best Known (89−45, 89, s)-Nets in Base 64
(89−45, 89, 513)-Net over F64 — Constructive and digital
Digital (44, 89, 513)-net over F64, using
- t-expansion [i] based on digital (28, 89, 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
(89−45, 89, 1424)-Net over F64 — Digital
Digital (44, 89, 1424)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6489, 1424, F64, 2, 45) (dual of [(1424, 2), 2759, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6489, 2049, F64, 2, 45) (dual of [(2049, 2), 4009, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6489, 4098, F64, 45) (dual of [4098, 4009, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(43) [i] based on
- linear OA(6489, 4096, F64, 45) (dual of [4096, 4007, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 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(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(44) ⊂ Ce(43) [i] based on
- OOA 2-folding [i] based on linear OA(6489, 4098, F64, 45) (dual of [4098, 4009, 46]-code), using
- discarding factors / shortening the dual code based on linear OOA(6489, 2049, F64, 2, 45) (dual of [(2049, 2), 4009, 46]-NRT-code), using
(89−45, 89, 2411184)-Net in Base 64 — Upper bound on s
There is no (44, 89, 2411185)-net in base 64, because
- 1 times m-reduction [i] would yield (44, 88, 2411185)-net in base 64, but
- 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]