Best Known (51−30, 51, s)-Nets in Base 81
(51−30, 51, 370)-Net over F81 — Constructive and digital
Digital (21, 51, 370)-net over F81, using
- t-expansion [i] based on digital (16, 51, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(51−30, 51, 374)-Net over F81 — Digital
Digital (21, 51, 374)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8151, 374, F81, 2, 30) (dual of [(374, 2), 697, 31]-NRT-code), using
- construction X applied to AG(2;F,707P) ⊂ AG(2;F,713P) [i] based on
- linear OOA(8146, 369, F81, 2, 30) (dual of [(369, 2), 692, 31]-NRT-code), using algebraic-geometric NRT-code AG(2;F,707P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- linear OOA(8140, 369, F81, 2, 24) (dual of [(369, 2), 698, 25]-NRT-code), using algebraic-geometric NRT-code AG(2;F,713P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370 (see above)
- linear OOA(815, 5, F81, 2, 5) (dual of [(5, 2), 5, 6]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(815, 81, F81, 2, 5) (dual of [(81, 2), 157, 6]-NRT-code), using
- Reed–Solomon NRT-code RS(2;157,81) [i]
- discarding factors / shortening the dual code based on linear OOA(815, 81, F81, 2, 5) (dual of [(81, 2), 157, 6]-NRT-code), using
- construction X applied to AG(2;F,707P) ⊂ AG(2;F,713P) [i] based on
(51−30, 51, 247464)-Net in Base 81 — Upper bound on s
There is no (21, 51, 247465)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 21 515237 828630 319815 775365 077975 738285 581765 583750 277573 254350 725059 681250 788243 349237 946464 798001 > 8151 [i]