Best Known (62−35, 62, s)-Nets in Base 64
(62−35, 62, 281)-Net over F64 — Constructive and digital
Digital (27, 62, 281)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 20, 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, 42, 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, 20, 104)-net over F64, using
(62−35, 62, 306)-Net in Base 64 — Constructive
(27, 62, 306)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 20, 129)-net in base 64, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- digital (7, 42, 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
- (3, 20, 129)-net in base 64, using
(62−35, 62, 442)-Net over F64 — Digital
Digital (27, 62, 442)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6462, 442, F64, 35) (dual of [442, 380, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 443, F64, 35) (dual of [443, 381, 36]-code), using
- 17 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0) [i] based on linear OA(6461, 425, F64, 35) (dual of [425, 364, 36]-code), using
- extended algebraic-geometric code AGe(F,389P) [i] based on function field F/F64 with g(F) = 26 and N(F) ≥ 425, using
- 17 step Varšamov–Edel lengthening with (ri) = (1, 16 times 0) [i] based on linear OA(6461, 425, F64, 35) (dual of [425, 364, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 443, F64, 35) (dual of [443, 381, 36]-code), using
(62−35, 62, 513)-Net in Base 64
(27, 62, 513)-net in base 64, using
- 14 times m-reduction [i] based on (27, 76, 513)-net in base 64, using
- base change [i] based on digital (8, 57, 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, 57, 513)-net over F256, using
(62−35, 62, 344820)-Net in Base 64 — Upper bound on s
There is no (27, 62, 344821)-net in base 64, because
- 1 times m-reduction [i] would yield (27, 61, 344821)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 150 311049 841050 299170 332068 601688 543985 230925 478825 849003 229731 773261 110855 887718 223769 071571 901315 235508 014596 > 6461 [i]