Best Known (85−49, 85, s)-Nets in Base 64
(85−49, 85, 513)-Net over F64 — Constructive and digital
Digital (36, 85, 513)-net over F64, using
- t-expansion [i] based on digital (28, 85, 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
(85−49, 85, 517)-Net over F64 — Digital
Digital (36, 85, 517)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6485, 517, F64, 3, 49) (dual of [(517, 3), 1466, 50]-NRT-code), using
- strength reduction [i] based on linear OOA(6485, 517, F64, 3, 50) (dual of [(517, 3), 1466, 51]-NRT-code), using
- construction X applied to AG(3;F,1485P) ⊂ AG(3;F,1493P) [i] based on
- linear OOA(6478, 512, F64, 3, 50) (dual of [(512, 3), 1458, 51]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1485P) [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(6470, 512, F64, 3, 42) (dual of [(512, 3), 1466, 43]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1493P) [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513 (see above)
- linear OOA(647, 5, F64, 3, 7) (dual of [(5, 3), 8, 8]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(647, 64, F64, 3, 7) (dual of [(64, 3), 185, 8]-NRT-code), using
- Reed–Solomon NRT-code RS(3;185,64) [i]
- discarding factors / shortening the dual code based on linear OOA(647, 64, F64, 3, 7) (dual of [(64, 3), 185, 8]-NRT-code), using
- linear OOA(6478, 512, F64, 3, 50) (dual of [(512, 3), 1458, 51]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1485P) [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- construction X applied to AG(3;F,1485P) ⊂ AG(3;F,1493P) [i] based on
- strength reduction [i] based on linear OOA(6485, 517, F64, 3, 50) (dual of [(517, 3), 1466, 51]-NRT-code), using
(85−49, 85, 326314)-Net in Base 64 — Upper bound on s
There is no (36, 85, 326315)-net in base 64, because
- 1 times m-reduction [i] would yield (36, 84, 326315)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 52 376432 392609 232385 422780 002760 930598 112065 071802 202213 167649 096545 050930 144555 014721 653456 707156 299179 716886 450721 478464 743560 827825 685180 022232 605666 > 6484 [i]