Best Known (76−36, 76, s)-Nets in Base 64
(76−36, 76, 513)-Net over F64 — Constructive and digital
Digital (40, 76, 513)-net over F64, using
- t-expansion [i] based on digital (28, 76, 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
(76−36, 76, 517)-Net in Base 64 — Constructive
(40, 76, 517)-net in base 64, using
- base change [i] based on digital (21, 57, 517)-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 (2, 38, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- digital (1, 19, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(76−36, 76, 2056)-Net over F64 — Digital
Digital (40, 76, 2056)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6476, 2056, F64, 36) (dual of [2056, 1980, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(6476, 4113, F64, 36) (dual of [4113, 4037, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(29) [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(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(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to Ce(35) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(6476, 4113, F64, 36) (dual of [4113, 4037, 37]-code), using
(76−36, 76, 5068529)-Net in Base 64 — Upper bound on s
There is no (40, 76, 5068530)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 186071 048558 758114 336207 926071 764969 304876 800870 456476 463219 817610 685572 091322 338587 735841 607205 099954 828354 975241 101022 067893 833321 192784 > 6476 [i]