Best Known (54−29, 54, s)-Nets in Base 64
(54−29, 54, 281)-Net over F64 — Constructive and digital
Digital (25, 54, 281)-net over F64, using
- 1 times m-reduction [i] based on digital (25, 55, 281)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 18, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (7, 37, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (3, 18, 104)-net over F64, using
- (u, u+v)-construction [i] based on
(54−29, 54, 337)-Net in Base 64 — Constructive
(25, 54, 337)-net in base 64, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, 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, 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, 30, 257)-net over F256, using
- 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
- digital (1, 15, 80)-net over F64, using
(54−29, 54, 587)-Net over F64 — Digital
Digital (25, 54, 587)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6454, 587, F64, 29) (dual of [587, 533, 30]-code), using
- construction X applied to C([18,46]) ⊂ C([19,46]) [i] based on
- linear OA(6454, 585, F64, 29) (dual of [585, 531, 30]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {18,19,…,46}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(6452, 585, F64, 28) (dual of [585, 533, 29]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {19,20,…,46}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([18,46]) ⊂ C([19,46]) [i] based on
(54−29, 54, 660380)-Net in Base 64 — Upper bound on s
There is no (25, 54, 660381)-net in base 64, because
- 1 times m-reduction [i] would yield (25, 53, 660381)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 534004 854636 047911 935065 520422 445692 853921 869104 862136 689655 010339 936000 833872 359714 929306 347320 > 6453 [i]