Best Known (54−31, 54, s)-Nets in Base 64
(54−31, 54, 257)-Net over F64 — Constructive and digital
Digital (23, 54, 257)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (7, 38, 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 (1, 16, 80)-net over F64, using
(54−31, 54, 288)-Net in Base 64 — Constructive
(23, 54, 288)-net in base 64, using
- t-expansion [i] based on (22, 54, 288)-net in base 64, using
- 37 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
- 37 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
(54−31, 54, 358)-Net over F64 — Digital
Digital (23, 54, 358)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6454, 358, F64, 31) (dual of [358, 304, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6454, 363, F64, 31) (dual of [363, 309, 32]-code), using
- 18 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 15 times 0) [i] based on linear OA(6451, 342, F64, 31) (dual of [342, 291, 32]-code), using
- extended algebraic-geometric code AGe(F,310P) [i] based on function field F/F64 with g(F) = 20 and N(F) ≥ 342, using
- 18 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 15 times 0) [i] based on linear OA(6451, 342, F64, 31) (dual of [342, 291, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(6454, 363, F64, 31) (dual of [363, 309, 32]-code), using
(54−31, 54, 513)-Net in Base 64
(23, 54, 513)-net in base 64, using
- 6 times m-reduction [i] based on (23, 60, 513)-net in base 64, using
- base change [i] based on digital (8, 45, 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, 45, 513)-net over F256, using
(54−31, 54, 245611)-Net in Base 64 — Upper bound on s
There is no (23, 54, 245612)-net in base 64, because
- 1 times m-reduction [i] would yield (23, 53, 245612)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 534003 775284 030230 685536 178405 035406 760583 071356 606295 584964 794695 034436 563722 408458 728343 748440 > 6453 [i]