Best Known (38−21, 38, s)-Nets in Base 64
(38−21, 38, 242)-Net over F64 — Constructive and digital
Digital (17, 38, 242)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (7, 28, 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 (0, 10, 65)-net over F64, using
(38−21, 38, 322)-Net in Base 64 — Constructive
(17, 38, 322)-net in base 64, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- (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
- digital (0, 10, 65)-net over F64, using
(38−21, 38, 367)-Net over F64 — Digital
Digital (17, 38, 367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6438, 367, F64, 21) (dual of [367, 329, 22]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (1, 48 times 0) [i] based on linear OA(6437, 317, F64, 21) (dual of [317, 280, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(6437, 316, F64, 21) (dual of [316, 279, 22]-code), using an extension Ce(20) of the narrow-sense BCH-code C(I) with length 315 | 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6436, 316, F64, 20) (dual of [316, 280, 21]-code), using an extension Ce(19) of the narrow-sense BCH-code C(I) with length 315 | 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(640, 1, F64, 0) (dual of [1, 1, 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 Ce(20) ⊂ Ce(19) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (1, 48 times 0) [i] based on linear OA(6437, 317, F64, 21) (dual of [317, 280, 22]-code), using
(38−21, 38, 346334)-Net in Base 64 — Upper bound on s
There is no (17, 38, 346335)-net in base 64, because
- 1 times m-reduction [i] would yield (17, 37, 346335)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 6 740074 675254 832842 997049 991399 035011 783467 842168 763813 124488 064735 > 6437 [i]