Best Known (78−36, 78, s)-Nets in Base 64
(78−36, 78, 513)-Net over F64 — Constructive and digital
Digital (42, 78, 513)-net over F64, using
- t-expansion [i] based on digital (28, 78, 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
(78−36, 78, 545)-Net in Base 64 — Constructive
(42, 78, 545)-net in base 64, using
- t-expansion [i] based on (41, 78, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 18, 257)-net over F256, using
- (17, 54, 288)-net in base 64, using
- 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 48, 288)-net over F128, using
- 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (6, 24, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(78−36, 78, 2630)-Net over F64 — Digital
Digital (42, 78, 2630)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6478, 2630, F64, 36) (dual of [2630, 2552, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(6478, 4119, F64, 36) (dual of [4119, 4041, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(27) [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(6455, 4096, F64, 28) (dual of [4096, 4041, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(647, 23, F64, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to Ce(35) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(6478, 4119, F64, 36) (dual of [4119, 4041, 37]-code), using
(78−36, 78, 8045794)-Net in Base 64 — Upper bound on s
There is no (42, 78, 8045795)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 762 146989 888121 678490 254353 586881 118554 303300 776866 337578 501488 939685 885437 660979 443195 266888 620077 596825 432448 238815 221169 787213 770796 176042 > 6478 [i]