Best Known (73−35, 73, s)-Nets in Base 64
(73−35, 73, 513)-Net over F64 — Constructive and digital
Digital (38, 73, 513)-net over F64, using
- t-expansion [i] based on digital (28, 73, 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
(73−35, 73, 516)-Net in Base 64 — Constructive
(38, 73, 516)-net in base 64, using
- 641 times duplication [i] based on (37, 72, 516)-net in base 64, using
- base change [i] based on digital (19, 54, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 36, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 18, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (19, 54, 516)-net over F256, using
(73−35, 73, 2050)-Net over F64 — Digital
Digital (38, 73, 2050)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6473, 2050, F64, 2, 35) (dual of [(2050, 2), 4027, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6473, 2055, F64, 2, 35) (dual of [(2055, 2), 4037, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6473, 4110, F64, 35) (dual of [4110, 4037, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(29) [i] based on
- linear OA(6469, 4096, F64, 35) (dual of [4096, 4027, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(34) ⊂ Ce(29) [i] based on
- OOA 2-folding [i] based on linear OA(6473, 4110, F64, 35) (dual of [4110, 4037, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(6473, 2055, F64, 2, 35) (dual of [(2055, 2), 4037, 36]-NRT-code), using
(73−35, 73, 5085192)-Net in Base 64 — Upper bound on s
There is no (38, 73, 5085193)-net in base 64, because
- 1 times m-reduction [i] would yield (38, 72, 5085193)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 11090 709261 741446 877987 320395 248027 114122 341952 307130 102545 811220 698990 512758 713584 593670 535917 227125 253417 743520 441216 341777 079296 > 6472 [i]