Best Known (56−28, 56, s)-Nets in Base 27
(56−28, 56, 164)-Net over F27 — Constructive and digital
Digital (28, 56, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 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 (7, 35, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 21, 82)-net over F27, using
(56−28, 56, 224)-Net in Base 27 — Constructive
(28, 56, 224)-net in base 27, using
- 4 times m-reduction [i] based on (28, 60, 224)-net in base 27, using
- base change [i] based on digital (13, 45, 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, 45, 224)-net over F81, using
(56−28, 56, 421)-Net over F27 — Digital
Digital (28, 56, 421)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2756, 421, F27, 28) (dual of [421, 365, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2756, 738, F27, 28) (dual of [738, 682, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- 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(2747, 729, F27, 24) (dual of [729, 682, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(273, 9, F27, 3) (dual of [9, 6, 4]-code or 9-arc in PG(2,27) or 9-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(2756, 738, F27, 28) (dual of [738, 682, 29]-code), using
(56−28, 56, 123570)-Net in Base 27 — Upper bound on s
There is no (28, 56, 123571)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 143 349663 855173 955890 966620 110600 749912 000112 108429 554955 339732 451458 797715 207261 > 2756 [i]