Best Known (42, 84, s)-Nets in Base 64
(42, 84, 513)-Net over F64 — Constructive and digital
Digital (42, 84, 513)-net over F64, using
- t-expansion [i] based on digital (28, 84, 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
(42, 84, 514)-Net in Base 64 — Constructive
(42, 84, 514)-net in base 64, using
- base change [i] based on digital (21, 63, 514)-net over F256, using
- (u, u+v)-construction [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
- digital (0, 42, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 21, 257)-net over F256, using
- (u, u+v)-construction [i] based on
(42, 84, 1514)-Net over F64 — Digital
Digital (42, 84, 1514)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6484, 1514, F64, 2, 42) (dual of [(1514, 2), 2944, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6484, 2050, F64, 2, 42) (dual of [(2050, 2), 4016, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6484, 4100, F64, 42) (dual of [4100, 4016, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(6484, 4101, F64, 42) (dual of [4101, 4017, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(39) [i] based on
- linear OA(6483, 4096, F64, 42) (dual of [4096, 4013, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(6479, 4096, F64, 40) (dual of [4096, 4017, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(41) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(6484, 4101, F64, 42) (dual of [4101, 4017, 43]-code), using
- OOA 2-folding [i] based on linear OA(6484, 4100, F64, 42) (dual of [4100, 4016, 43]-code), using
- discarding factors / shortening the dual code based on linear OOA(6484, 2050, F64, 2, 42) (dual of [(2050, 2), 4016, 43]-NRT-code), using
(42, 84, 2311372)-Net in Base 64 — Upper bound on s
There is no (42, 84, 2311373)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 52 374411 013864 001584 298517 126579 803366 098184 081014 413755 534796 830422 010351 780636 624684 340248 499500 530745 060102 387865 049925 202293 135022 769356 770192 530720 > 6484 [i]