Best Known (65−37, 65, s)-Nets in Base 27
(65−37, 65, 140)-Net over F27 — Constructive and digital
Digital (28, 65, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 22, 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, 43, 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, 22, 64)-net over F27, using
(65−37, 65, 172)-Net in Base 27 — Constructive
(28, 65, 172)-net in base 27, using
- 19 times m-reduction [i] based on (28, 84, 172)-net in base 27, using
- base change [i] based on digital (7, 63, 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, 63, 172)-net over F81, using
(65−37, 65, 211)-Net over F27 — Digital
Digital (28, 65, 211)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2765, 211, F27, 2, 37) (dual of [(211, 2), 357, 38]-NRT-code), using
- construction X applied to AG(2;F,376P) ⊂ AG(2;F,381P) [i] based on
- linear OOA(2761, 207, F27, 2, 37) (dual of [(207, 2), 353, 38]-NRT-code), using algebraic-geometric NRT-code AG(2;F,376P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208, using
- linear OOA(2756, 207, F27, 2, 32) (dual of [(207, 2), 358, 33]-NRT-code), using algebraic-geometric NRT-code AG(2;F,381P) [i] based on function field F/F27 with g(F) = 24 and N(F) ≥ 208 (see above)
- linear OOA(274, 4, F27, 2, 4) (dual of [(4, 2), 4, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(274, 27, F27, 2, 4) (dual of [(27, 2), 50, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(2;50,27) [i]
- discarding factors / shortening the dual code based on linear OOA(274, 27, F27, 2, 4) (dual of [(27, 2), 50, 5]-NRT-code), using
- construction X applied to AG(2;F,376P) ⊂ AG(2;F,381P) [i] based on
(65−37, 65, 244)-Net in Base 27
(28, 65, 244)-net in base 27, using
- 11 times m-reduction [i] based on (28, 76, 244)-net in base 27, using
- base change [i] based on digital (9, 57, 244)-net over F81, using
- net from sequence [i] based on digital (9, 243)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 9 and N(F) ≥ 244, using
- net from sequence [i] based on digital (9, 243)-sequence over F81, using
- base change [i] based on digital (9, 57, 244)-net over F81, using
(65−37, 65, 35672)-Net in Base 27 — Upper bound on s
There is no (28, 65, 35673)-net in base 27, because
- 1 times m-reduction [i] would yield (28, 64, 35673)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 40 486166 245665 764146 282242 147282 967835 285455 832735 594611 126313 275874 662451 910185 122375 499913 > 2764 [i]