Best Known (59−31, 59, s)-Nets in Base 27
(59−31, 59, 158)-Net over F27 — Constructive and digital
Digital (28, 59, 158)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 21, 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, 38, 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, 21, 76)-net over F27, using
(59−31, 59, 224)-Net in Base 27 — Constructive
(28, 59, 224)-net in base 27, using
- 1 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
(59−31, 59, 343)-Net over F27 — Digital
Digital (28, 59, 343)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2759, 343, F27, 2, 31) (dual of [(343, 2), 627, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2759, 367, F27, 2, 31) (dual of [(367, 2), 675, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2759, 734, F27, 31) (dual of [734, 675, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2758, 729, F27, 31) (dual of [729, 671, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(2759, 734, F27, 31) (dual of [734, 675, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2759, 367, F27, 2, 31) (dual of [(367, 2), 675, 32]-NRT-code), using
(59−31, 59, 84598)-Net in Base 27 — Upper bound on s
There is no (28, 59, 84599)-net in base 27, because
- 1 times m-reduction [i] would yield (28, 58, 84599)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 104511 110394 757300 626037 479917 827888 438944 688337 786322 056441 132044 396521 876166 788891 > 2758 [i]