Best Known (18, 18+18, s)-Nets in Base 64
(18, 18+18, 455)-Net over F64 — Constructive and digital
Digital (18, 36, 455)-net over F64, using
- 1 times m-reduction [i] based on digital (18, 37, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 455, F64, 19, 19) (dual of [(455, 19), 8608, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using
- net defined by OOA [i] based on linear OOA(6437, 455, F64, 19, 19) (dual of [(455, 19), 8608, 20]-NRT-code), using
(18, 18+18, 514)-Net in Base 64 — Constructive
(18, 36, 514)-net in base 64, using
- base change [i] based on digital (9, 27, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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, 18, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 9, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(18, 18+18, 1367)-Net over F64 — Digital
Digital (18, 36, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6436, 1367, F64, 3, 18) (dual of [(1367, 3), 4065, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6436, 4101, F64, 18) (dual of [4101, 4065, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(6436, 4101, F64, 18) (dual of [4101, 4065, 19]-code), using
(18, 18+18, 1104407)-Net in Base 64 — Upper bound on s
There is no (18, 36, 1104408)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 105313 117540 740613 715489 659059 362596 511919 372910 210320 115967 035162 > 6436 [i]