Best Known (34, 65, s)-Nets in Base 64
(34, 65, 513)-Net over F64 — Constructive and digital
Digital (34, 65, 513)-net over F64, using
- t-expansion [i] based on digital (28, 65, 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
(34, 65, 516)-Net in Base 64 — Constructive
(34, 65, 516)-net in base 64, using
- 641 times duplication [i] based on (33, 64, 516)-net in base 64, using
- base change [i] based on digital (17, 48, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 32, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 16, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (17, 48, 516)-net over F256, using
(34, 65, 2055)-Net over F64 — Digital
Digital (34, 65, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6465, 2055, F64, 2, 31) (dual of [(2055, 2), 4045, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6465, 4110, F64, 31) (dual of [4110, 4045, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(6461, 4096, F64, 31) (dual of [4096, 4035, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(30) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(6465, 4110, F64, 31) (dual of [4110, 4045, 32]-code), using
(34, 65, 5185532)-Net in Base 64 — Upper bound on s
There is no (34, 65, 5185533)-net in base 64, because
- 1 times m-reduction [i] would yield (34, 64, 5185533)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 39 402044 720880 281284 267218 425641 879602 339860 607911 516317 408576 112943 427602 293983 434828 034578 457221 829907 235796 215088 > 6464 [i]