Best Known (75−36, 75, s)-Nets in Base 64
(75−36, 75, 513)-Net over F64 — Constructive and digital
Digital (39, 75, 513)-net over F64, using
- t-expansion [i] based on digital (28, 75, 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
(75−36, 75, 516)-Net in Base 64 — Constructive
(39, 75, 516)-net in base 64, using
- 1 times m-reduction [i] based on (39, 76, 516)-net in base 64, using
- base change [i] based on digital (20, 57, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 19, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 38, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 19, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (20, 57, 516)-net over F256, using
(75−36, 75, 2052)-Net over F64 — Digital
Digital (39, 75, 2052)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6475, 2052, F64, 2, 36) (dual of [(2052, 2), 4029, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6475, 2055, F64, 2, 36) (dual of [(2055, 2), 4035, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6475, 4110, F64, 36) (dual of [4110, 4035, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(30) [i] based on
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 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(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(35) ⊂ Ce(30) [i] based on
- OOA 2-folding [i] based on linear OA(6475, 4110, F64, 36) (dual of [4110, 4035, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(6475, 2055, F64, 2, 36) (dual of [(2055, 2), 4035, 37]-NRT-code), using
(75−36, 75, 4022892)-Net in Base 64 — Upper bound on s
There is no (39, 75, 4022893)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 2907 358644 509425 407228 382308 452829 790610 864832 609624 438146 353684 736947 519158 824925 044471 853284 006127 070347 620985 660640 093317 564580 644460 > 6475 [i]