Best Known (51−24, 51, s)-Nets in Base 64
(51−24, 51, 354)-Net over F64 — Constructive and digital
Digital (27, 51, 354)-net over F64, using
- 2 times m-reduction [i] based on digital (27, 53, 354)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (7, 33, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64 (see above)
- digital (7, 20, 177)-net over F64, using
- (u, u+v)-construction [i] based on
(51−24, 51, 516)-Net in Base 64 — Constructive
(27, 51, 516)-net in base 64, using
- 1 times m-reduction [i] based on (27, 52, 516)-net in base 64, using
- base change [i] based on digital (14, 39, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 26, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 13, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (14, 39, 516)-net over F256, using
(51−24, 51, 2055)-Net over F64 — Digital
Digital (27, 51, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6451, 2055, F64, 2, 24) (dual of [(2055, 2), 4059, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6451, 4110, F64, 24) (dual of [4110, 4059, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(6451, 4110, F64, 24) (dual of [4110, 4059, 25]-code), using
(51−24, 51, 3983686)-Net in Base 64 — Upper bound on s
There is no (27, 51, 3983687)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 130 370573 021025 227988 412585 300706 779967 630485 391109 694772 492312 208011 485250 151134 411354 878840 > 6451 [i]