Best Known (53−28, 53, s)-Nets in Base 27
(53−28, 53, 146)-Net over F27 — Constructive and digital
Digital (25, 53, 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, 35, 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
(53−28, 53, 172)-Net in Base 27 — Constructive
(25, 53, 172)-net in base 27, using
- 19 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
(53−28, 53, 313)-Net over F27 — Digital
Digital (25, 53, 313)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2753, 313, F27, 2, 28) (dual of [(313, 2), 573, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2753, 364, F27, 2, 28) (dual of [(364, 2), 675, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2753, 728, F27, 28) (dual of [728, 675, 29]-code), using
- 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]
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 2-folding [i] based on linear OA(2753, 728, F27, 28) (dual of [728, 675, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(2753, 364, F27, 2, 28) (dual of [(364, 2), 675, 29]-NRT-code), using
(53−28, 53, 60977)-Net in Base 27 — Upper bound on s
There is no (25, 53, 60978)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 7283 577220 992361 251819 977509 356091 233507 041135 820595 227878 198783 170593 099501 > 2753 [i]