Best Known (27, 27+29, s)-Nets in Base 27
(27, 27+29, 158)-Net over F27 — Constructive and digital
Digital (27, 56, 158)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 20, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-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 (6, 20, 76)-net over F27, using
(27, 27+29, 224)-Net in Base 27 — Constructive
(27, 56, 224)-net in base 27, using
- base change [i] based on digital (13, 42, 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
(27, 27+29, 367)-Net over F27 — Digital
Digital (27, 56, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2756, 367, F27, 2, 29) (dual of [(367, 2), 678, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2756, 734, F27, 29) (dual of [734, 678, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [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(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(272, 5, F27, 2) (dual of [5, 3, 3]-code or 5-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(2756, 734, F27, 29) (dual of [734, 678, 30]-code), using
(27, 27+29, 97649)-Net in Base 27 — Upper bound on s
There is no (27, 56, 97650)-net in base 27, because
- 1 times m-reduction [i] would yield (27, 55, 97650)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 5 309678 297239 053837 495454 293828 445356 259418 295654 345480 218195 227921 497209 800301 > 2755 [i]