Best Known (65−33, 65, s)-Nets in Base 27
(65−33, 65, 170)-Net over F27 — Constructive and digital
Digital (32, 65, 170)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 23, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (9, 42, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (7, 23, 82)-net over F27, using
(65−33, 65, 224)-Net in Base 27 — Constructive
(32, 65, 224)-net in base 27, using
- 11 times m-reduction [i] based on (32, 76, 224)-net in base 27, using
- base change [i] based on digital (13, 57, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 57, 224)-net over F81, using
(65−33, 65, 417)-Net over F27 — Digital
Digital (32, 65, 417)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2765, 417, F27, 33) (dual of [417, 352, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2765, 728, F27, 33) (dual of [728, 663, 34]-code), using
(65−33, 65, 138993)-Net in Base 27 — Upper bound on s
There is no (32, 65, 138994)-net in base 27, because
- 1 times m-reduction [i] would yield (32, 64, 138994)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 40 485221 629897 814790 051023 460928 461839 966735 033571 145117 244350 094308 957883 004064 478318 815905 > 2764 [i]