Best Known (77−36, 77, s)-Nets in Base 64
(77−36, 77, 513)-Net over F64 — Constructive and digital
Digital (41, 77, 513)-net over F64, using
- t-expansion [i] based on digital (28, 77, 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
(77−36, 77, 545)-Net in Base 64 — Constructive
(41, 77, 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, 53, 288)-net in base 64, using
- 3 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
- 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (6, 24, 257)-net in base 64, using
(77−36, 77, 2325)-Net over F64 — Digital
Digital (41, 77, 2325)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6477, 2325, F64, 36) (dual of [2325, 2248, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(6477, 4116, F64, 36) (dual of [4116, 4039, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(28) [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(6457, 4096, F64, 29) (dual of [4096, 4039, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(646, 20, F64, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(35) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(6477, 4116, F64, 36) (dual of [4116, 4039, 37]-code), using
(77−36, 77, 6385949)-Net in Base 64 — Upper bound on s
There is no (41, 77, 6385950)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 11 908549 705323 358300 495815 874811 247229 974869 774763 459698 487611 451590 369013 431655 651803 888231 029009 578430 511487 303072 372665 260632 806156 280586 > 6477 [i]