Best Known (39, 77, s)-Nets in Base 64
(39, 77, 513)-Net over F64 — Constructive and digital
Digital (39, 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
(39, 77, 514)-Net in Base 64 — Constructive
(39, 77, 514)-net in base 64, using
- 1 times m-reduction [i] based on (39, 78, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 26, 257)-net in base 64, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (13, 52, 257)-net in base 64, using
- base change [i] based on digital (0, 39, 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, 39, 257)-net over F256, using
- (7, 26, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(39, 77, 1621)-Net over F64 — Digital
Digital (39, 77, 1621)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6477, 1621, F64, 2, 38) (dual of [(1621, 2), 3165, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6477, 2052, F64, 2, 38) (dual of [(2052, 2), 4027, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6477, 4104, F64, 38) (dual of [4104, 4027, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(34) [i] based on
- linear OA(6475, 4096, F64, 38) (dual of [4096, 4021, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(6469, 4096, F64, 35) (dual of [4096, 4027, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(37) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(6477, 4104, F64, 38) (dual of [4104, 4027, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(6477, 2052, F64, 2, 38) (dual of [(2052, 2), 4027, 39]-NRT-code), using
(39, 77, 2628181)-Net in Base 64 — Upper bound on s
There is no (39, 77, 2628182)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 11 908534 432492 451912 313992 522678 925554 272038 878804 987017 713251 835788 185483 679618 015048 641397 905486 725702 651457 114709 499573 704938 928915 628982 > 6477 [i]