Best Known (23−13, 23, s)-Nets in Base 64
(23−13, 23, 195)-Net over F64 — Constructive and digital
Digital (10, 23, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 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 (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 13, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 4, 65)-net over F64, using
(23−13, 23, 261)-Net in Base 64 — Constructive
(10, 23, 261)-net in base 64, using
- 1 times m-reduction [i] based on (10, 24, 261)-net in base 64, using
- base change [i] based on digital (4, 18, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- base change [i] based on digital (4, 18, 261)-net over F256, using
(23−13, 23, 315)-Net over F64 — Digital
Digital (10, 23, 315)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6423, 315, F64, 13) (dual of [315, 292, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 315 | 642−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
(23−13, 23, 321)-Net in Base 64
(10, 23, 321)-net in base 64, using
- 9 times m-reduction [i] based on (10, 32, 321)-net in base 64, using
- base change [i] based on digital (2, 24, 321)-net over F256, using
- net from sequence [i] based on digital (2, 320)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- net from sequence [i] based on digital (2, 320)-sequence over F256, using
- base change [i] based on digital (2, 24, 321)-net over F256, using
(23−13, 23, 199313)-Net in Base 64 — Upper bound on s
There is no (10, 23, 199314)-net in base 64, because
- 1 times m-reduction [i] would yield (10, 22, 199314)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 5444 669961 864660 747052 208719 070316 726040 > 6422 [i]