Best Known (24, 24+34, s)-Nets in Base 64
(24, 24+34, 242)-Net over F64 — Constructive and digital
Digital (24, 58, 242)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 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, 41, 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, 17, 65)-net over F64, using
(24, 24+34, 288)-Net in Base 64 — Constructive
(24, 58, 288)-net in base 64, using
- t-expansion [i] based on (22, 58, 288)-net in base 64, using
- 33 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 78, 288)-net over F128, using
- 33 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
(24, 24+34, 345)-Net over F64 — Digital
Digital (24, 58, 345)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6458, 345, F64, 2, 34) (dual of [(345, 2), 632, 35]-NRT-code), using
- construction X applied to AG(2;F,647P) ⊂ AG(2;F,652P) [i] based on
- linear OOA(6454, 341, F64, 2, 34) (dual of [(341, 2), 628, 35]-NRT-code), using algebraic-geometric NRT-code AG(2;F,647P) [i] based on function field F/F64 with g(F) = 20 and N(F) ≥ 342, using
- linear OOA(6449, 341, F64, 2, 29) (dual of [(341, 2), 633, 30]-NRT-code), using algebraic-geometric NRT-code AG(2;F,652P) [i] based on function field F/F64 with g(F) = 20 and N(F) ≥ 342 (see above)
- linear OOA(644, 4, F64, 2, 4) (dual of [(4, 2), 4, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(644, 64, F64, 2, 4) (dual of [(64, 2), 124, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(2;124,64) [i]
- discarding factors / shortening the dual code based on linear OOA(644, 64, F64, 2, 4) (dual of [(64, 2), 124, 5]-NRT-code), using
- construction X applied to AG(2;F,647P) ⊂ AG(2;F,652P) [i] based on
(24, 24+34, 513)-Net in Base 64
(24, 58, 513)-net in base 64, using
- 6 times m-reduction [i] based on (24, 64, 513)-net in base 64, using
- base change [i] based on digital (8, 48, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 48, 513)-net over F256, using
(24, 24+34, 165517)-Net in Base 64 — Upper bound on s
There is no (24, 58, 165518)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 573 401206 446183 148757 355264 059446 036441 874964 505081 899799 262186 006644 421416 772819 864556 904995 428874 436216 > 6458 [i]