Best Known (29, 29+39, s)-Nets in Base 27
(29, 29+39, 140)-Net over F27 — Constructive and digital
Digital (29, 68, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 23, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (6, 45, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (4, 23, 64)-net over F27, using
(29, 29+39, 172)-Net in Base 27 — Constructive
(29, 68, 172)-net in base 27, using
- 20 times m-reduction [i] based on (29, 88, 172)-net in base 27, using
- base change [i] based on digital (7, 66, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 66, 172)-net over F81, using
(29, 29+39, 212)-Net over F27 — Digital
Digital (29, 68, 212)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2768, 212, F27, 2, 39) (dual of [(212, 2), 356, 40]-NRT-code), using
- construction X applied to AG(2;F,374P) ⊂ AG(2;F,380P) [i] based on
- linear OOA(2763, 207, F27, 2, 39) (dual of [(207, 2), 351, 40]-NRT-code), using algebraic-geometric NRT-code AG(2;F,374P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208, using
- linear OOA(2757, 207, F27, 2, 33) (dual of [(207, 2), 357, 34]-NRT-code), using algebraic-geometric NRT-code AG(2;F,380P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208 (see above)
- linear OOA(275, 5, F27, 2, 5) (dual of [(5, 2), 5, 6]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(275, 27, F27, 2, 5) (dual of [(27, 2), 49, 6]-NRT-code), using
- Reed–Solomon NRT-code RS(2;49,27) [i]
- discarding factors / shortening the dual code based on linear OOA(275, 27, F27, 2, 5) (dual of [(27, 2), 49, 6]-NRT-code), using
- construction X applied to AG(2;F,374P) ⊂ AG(2;F,380P) [i] based on
(29, 29+39, 298)-Net in Base 27
(29, 68, 298)-net in base 27, using
- base change [i] based on digital (12, 51, 298)-net over F81, using
- net from sequence [i] based on digital (12, 297)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 12 and N(F) ≥ 298, using
- net from sequence [i] based on digital (12, 297)-sequence over F81, using
(29, 29+39, 34006)-Net in Base 27 — Upper bound on s
There is no (29, 68, 34007)-net in base 27, because
- 1 times m-reduction [i] would yield (29, 67, 34007)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 797182 426928 156155 796551 486965 346394 768408 826161 366106 082347 581657 781678 311113 652954 507586 411315 > 2767 [i]