Best Known (74−43, 74, s)-Nets in Base 27
(74−43, 74, 140)-Net over F27 — Constructive and digital
Digital (31, 74, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 25, 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, 49, 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, 25, 64)-net over F27, using
(74−43, 74, 172)-Net in Base 27 — Constructive
(31, 74, 172)-net in base 27, using
- 22 times m-reduction [i] based on (31, 96, 172)-net in base 27, using
- base change [i] based on digital (7, 72, 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, 72, 172)-net over F81, using
(74−43, 74, 209)-Net over F27 — Digital
Digital (31, 74, 209)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2774, 209, F27, 2, 43) (dual of [(209, 2), 344, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2774, 214, F27, 2, 43) (dual of [(214, 2), 354, 44]-NRT-code), using
- construction X applied to AG(2;F,370P) ⊂ AG(2;F,378P) [i] based on
- linear OOA(2767, 207, F27, 2, 43) (dual of [(207, 2), 347, 44]-NRT-code), using algebraic-geometric NRT-code AG(2;F,370P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208, using
- linear OOA(2759, 207, F27, 2, 35) (dual of [(207, 2), 355, 36]-NRT-code), using algebraic-geometric NRT-code AG(2;F,378P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208 (see above)
- linear OOA(277, 7, F27, 2, 7) (dual of [(7, 2), 7, 8]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(277, 27, F27, 2, 7) (dual of [(27, 2), 47, 8]-NRT-code), using
- Reed–Solomon NRT-code RS(2;47,27) [i]
- discarding factors / shortening the dual code based on linear OOA(277, 27, F27, 2, 7) (dual of [(27, 2), 47, 8]-NRT-code), using
- construction X applied to AG(2;F,370P) ⊂ AG(2;F,378P) [i] based on
- discarding factors / shortening the dual code based on linear OOA(2774, 214, F27, 2, 43) (dual of [(214, 2), 354, 44]-NRT-code), using
(74−43, 74, 298)-Net in Base 27
(31, 74, 298)-net in base 27, using
- 2 times m-reduction [i] based on (31, 76, 298)-net in base 27, using
- base change [i] based on digital (12, 57, 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
- base change [i] based on digital (12, 57, 298)-net over F81, using
(74−43, 74, 31554)-Net in Base 27 — Upper bound on s
There is no (31, 74, 31555)-net in base 27, because
- 1 times m-reduction [i] would yield (31, 73, 31555)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 308 781437 348436 245977 015267 623302 800827 871064 707320 724417 865005 684871 893393 639467 646373 763552 867146 477511 > 2773 [i]