Best Known (34, 68, s)-Nets in Base 27
(34, 68, 176)-Net over F27 — Constructive and digital
Digital (34, 68, 176)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 24, 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 (10, 44, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (7, 24, 82)-net over F27, using
(34, 68, 370)-Net in Base 27 — Constructive
(34, 68, 370)-net in base 27, using
- 4 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
- base change [i] based on digital (16, 54, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 54, 370)-net over F81, using
(34, 68, 474)-Net over F27 — Digital
Digital (34, 68, 474)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2768, 474, F27, 34) (dual of [474, 406, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2768, 743, F27, 34) (dual of [743, 675, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- linear OA(2764, 729, F27, 34) (dual of [729, 665, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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(274, 14, F27, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2768, 743, F27, 34) (dual of [743, 675, 35]-code), using
(34, 68, 146690)-Net in Base 27 — Upper bound on s
There is no (34, 68, 146691)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 21 516563 458465 510015 264888 037270 353604 537089 445886 078865 837412 089623 384513 284984 480320 475869 449007 > 2768 [i]