Best Known (24, 59, s)-Nets in Base 81
(24, 59, 370)-Net over F81 — Constructive and digital
Digital (24, 59, 370)-net over F81, using
- t-expansion [i] based on digital (16, 59, 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
(24, 59, 377)-Net over F81 — Digital
Digital (24, 59, 377)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8159, 377, F81, 2, 35) (dual of [(377, 2), 695, 36]-NRT-code), using
- construction X applied to AG(2;F,702P) ⊂ AG(2;F,711P) [i] based on
- linear OOA(8151, 369, F81, 2, 35) (dual of [(369, 2), 687, 36]-NRT-code), using algebraic-geometric NRT-code AG(2;F,702P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- linear OOA(8142, 369, F81, 2, 26) (dual of [(369, 2), 696, 27]-NRT-code), using algebraic-geometric NRT-code AG(2;F,711P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370 (see above)
- linear OOA(818, 8, F81, 2, 8) (dual of [(8, 2), 8, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(818, 81, F81, 2, 8) (dual of [(81, 2), 154, 9]-NRT-code), using
- Reed–Solomon NRT-code RS(2;154,81) [i]
- discarding factors / shortening the dual code based on linear OOA(818, 81, F81, 2, 8) (dual of [(81, 2), 154, 9]-NRT-code), using
- construction X applied to AG(2;F,702P) ⊂ AG(2;F,711P) [i] based on
(24, 59, 291168)-Net in Base 81 — Upper bound on s
There is no (24, 59, 291169)-net in base 81, because
- 1 times m-reduction [i] would yield (24, 58, 291169)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 492 194541 120068 923043 339502 822332 011583 619959 754508 142766 921228 773088 729860 489303 324434 014602 263671 764502 485841 > 8158 [i]