Best Known (29, 29+28, s)-Nets in Base 64
(29, 29+28, 513)-Net over F64 — Constructive and digital
Digital (29, 57, 513)-net over F64, using
- t-expansion [i] based on digital (28, 57, 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
(29, 29+28, 514)-Net in Base 64 — Constructive
(29, 57, 514)-net in base 64, using
- 1 times m-reduction [i] based on (29, 58, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 19, 257)-net in base 64, using
- 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- base change [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- 1 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
- (10, 39, 257)-net in base 64, using
- 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
- base change [i] based on digital (0, 30, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 30, 257)-net over F256, using
- 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
- (5, 19, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(29, 29+28, 1508)-Net over F64 — Digital
Digital (29, 57, 1508)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6457, 1508, F64, 2, 28) (dual of [(1508, 2), 2959, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6457, 2052, F64, 2, 28) (dual of [(2052, 2), 4047, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6457, 4104, F64, 28) (dual of [4104, 4047, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- 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(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(6457, 4104, F64, 28) (dual of [4104, 4047, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(6457, 2052, F64, 2, 28) (dual of [(2052, 2), 4047, 29]-NRT-code), using
(29, 29+28, 2166948)-Net in Base 64 — Upper bound on s
There is no (29, 57, 2166949)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 8 959006 750496 207441 052443 370082 135541 759832 890746 638215 646631 724232 174500 117466 277386 184292 254107 151024 > 6457 [i]