Best Known (79−45, 79, s)-Nets in Base 64
(79−45, 79, 513)-Net over F64 — Constructive and digital
Digital (34, 79, 513)-net over F64, using
- t-expansion [i] based on digital (28, 79, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(79−45, 79, 516)-Net over F64 — Digital
Digital (34, 79, 516)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6479, 516, F64, 2, 45) (dual of [(516, 2), 953, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6479, 518, F64, 2, 45) (dual of [(518, 2), 957, 46]-NRT-code), using
- construction X applied to AG(2;F,978P) ⊂ AG(2;F,985P) [i] based on
- linear OOA(6473, 512, F64, 2, 45) (dual of [(512, 2), 951, 46]-NRT-code), using algebraic-geometric NRT-code AG(2;F,978P) [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- linear OOA(6466, 512, F64, 2, 38) (dual of [(512, 2), 958, 39]-NRT-code), using algebraic-geometric NRT-code AG(2;F,985P) [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513 (see above)
- linear OOA(646, 6, F64, 2, 6) (dual of [(6, 2), 6, 7]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(646, 64, F64, 2, 6) (dual of [(64, 2), 122, 7]-NRT-code), using
- Reed–Solomon NRT-code RS(2;122,64) [i]
- discarding factors / shortening the dual code based on linear OOA(646, 64, F64, 2, 6) (dual of [(64, 2), 122, 7]-NRT-code), using
- linear OOA(6473, 512, F64, 2, 45) (dual of [(512, 2), 951, 46]-NRT-code), using algebraic-geometric NRT-code AG(2;F,978P) [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- construction X applied to AG(2;F,978P) ⊂ AG(2;F,985P) [i] based on
- discarding factors / shortening the dual code based on linear OOA(6479, 518, F64, 2, 45) (dual of [(518, 2), 957, 46]-NRT-code), using
(79−45, 79, 364106)-Net in Base 64 — Upper bound on s
There is no (34, 79, 364107)-net in base 64, because
- 1 times m-reduction [i] would yield (34, 78, 364107)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 762 164657 803530 167186 696445 699063 822092 773712 370296 190789 550359 616627 648208 692896 385674 054510 633652 263612 396278 980839 895265 530092 519311 959480 > 6478 [i]