Best Known (40, 40+38, s)-Nets in Base 64
(40, 40+38, 513)-Net over F64 — Constructive and digital
Digital (40, 78, 513)-net over F64, using
- t-expansion [i] based on digital (28, 78, 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
(40, 40+38, 515)-Net in Base 64 — Constructive
(40, 78, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 26, 257)-net in base 64, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 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
- base change [i] based on digital (0, 21, 257)-net over F256, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (14, 52, 258)-net in base 64, using
- base change [i] based on digital (1, 39, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 39, 258)-net over F256, using
- (7, 26, 257)-net in base 64, using
(40, 40+38, 1828)-Net over F64 — Digital
Digital (40, 78, 1828)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6478, 1828, F64, 2, 38) (dual of [(1828, 2), 3578, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6478, 2053, F64, 2, 38) (dual of [(2053, 2), 4028, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6478, 4106, F64, 38) (dual of [4106, 4028, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(6478, 4107, F64, 38) (dual of [4107, 4029, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(6475, 4096, F64, 38) (dual of [4096, 4021, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(6478, 4107, F64, 38) (dual of [4107, 4029, 39]-code), using
- OOA 2-folding [i] based on linear OA(6478, 4106, F64, 38) (dual of [4106, 4028, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(6478, 2053, F64, 2, 38) (dual of [(2053, 2), 4028, 39]-NRT-code), using
(40, 40+38, 3271280)-Net in Base 64 — Upper bound on s
There is no (40, 78, 3271281)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 762 146490 572564 939077 256531 711063 201317 030224 977449 731079 854621 463590 041836 721894 489327 682020 354489 368258 264933 904822 451259 799280 969293 830048 > 6478 [i]