Best Known (49−29, 49, s)-Nets in Base 81
(49−29, 49, 370)-Net over F81 — Constructive and digital
Digital (20, 49, 370)-net over F81, using
- t-expansion [i] based on digital (16, 49, 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
(49−29, 49, 372)-Net over F81 — Digital
Digital (20, 49, 372)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8149, 372, F81, 3, 29) (dual of [(372, 3), 1067, 30]-NRT-code), using
- construction X applied to AG(3;F,1077P) ⊂ AG(3;F,1082P) [i] based on
- linear OOA(8145, 369, F81, 3, 29) (dual of [(369, 3), 1062, 30]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1077P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- linear OOA(8140, 369, F81, 3, 24) (dual of [(369, 3), 1067, 25]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1082P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370 (see above)
- linear OOA(814, 3, F81, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(814, 81, F81, 3, 4) (dual of [(81, 3), 239, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(3;239,81) [i]
- discarding factors / shortening the dual code based on linear OOA(814, 81, F81, 3, 4) (dual of [(81, 3), 239, 5]-NRT-code), using
- construction X applied to AG(3;F,1077P) ⊂ AG(3;F,1082P) [i] based on
(49−29, 49, 264079)-Net in Base 81 — Upper bound on s
There is no (20, 49, 264080)-net in base 81, because
- 1 times m-reduction [i] would yield (20, 48, 264080)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 40 485760 848939 569619 373079 133536 352504 857407 486649 995913 833912 684047 096228 599554 980174 425601 > 8148 [i]