Best Known (25, 25+29, s)-Nets in Base 27
(25, 25+29, 146)-Net over F27 — Constructive and digital
Digital (25, 54, 146)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 18, 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 (7, 36, 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 (4, 18, 64)-net over F27, using
(25, 25+29, 172)-Net in Base 27 — Constructive
(25, 54, 172)-net in base 27, using
- 18 times m-reduction [i] based on (25, 72, 172)-net in base 27, using
- base change [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(25, 25+29, 283)-Net over F27 — Digital
Digital (25, 54, 283)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2754, 283, F27, 2, 29) (dual of [(283, 2), 512, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2754, 365, F27, 2, 29) (dual of [(365, 2), 676, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2754, 730, F27, 29) (dual of [730, 676, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(2754, 729, F27, 29) (dual of [729, 675, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2753, 729, F27, 28) (dual of [729, 676, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(270, 1, F27, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(2754, 730, F27, 29) (dual of [730, 676, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2754, 365, F27, 2, 29) (dual of [(365, 2), 676, 30]-NRT-code), using
(25, 25+29, 60977)-Net in Base 27 — Upper bound on s
There is no (25, 54, 60978)-net in base 27, because
- 1 times m-reduction [i] would yield (25, 53, 60978)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 7283 577220 992361 251819 977509 356091 233507 041135 820595 227878 198783 170593 099501 > 2753 [i]