Best Known (28, 54, s)-Nets in Base 27
(28, 54, 166)-Net over F27 — Constructive and digital
Digital (28, 54, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 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 (8, 34, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (7, 20, 82)-net over F27, using
(28, 54, 224)-Net in Base 27 — Constructive
(28, 54, 224)-net in base 27, using
- 6 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
(28, 54, 535)-Net over F27 — Digital
Digital (28, 54, 535)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2754, 535, F27, 26) (dual of [535, 481, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2754, 740, F27, 26) (dual of [740, 686, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(2751, 729, F27, 26) (dual of [729, 678, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2743, 729, F27, 22) (dual of [729, 686, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(273, 11, F27, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-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(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(2754, 740, F27, 26) (dual of [740, 686, 27]-code), using
(28, 54, 192345)-Net in Base 27 — Upper bound on s
There is no (28, 54, 192346)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 196627 864797 137602 764593 109010 570703 153975 491140 309293 487390 701118 011565 936813 > 2754 [i]