Best Known (29, 29+32, s)-Nets in Base 27
(29, 29+32, 158)-Net over F27 — Constructive and digital
Digital (29, 61, 158)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 22, 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, 39, 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, 22, 76)-net over F27, using
(29, 29+32, 224)-Net in Base 27 — Constructive
(29, 61, 224)-net in base 27, using
- 3 times m-reduction [i] based on (29, 64, 224)-net in base 27, using
- base change [i] based on digital (13, 48, 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, 48, 224)-net over F81, using
(29, 29+32, 352)-Net over F27 — Digital
Digital (29, 61, 352)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2761, 352, F27, 2, 32) (dual of [(352, 2), 643, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2761, 367, F27, 2, 32) (dual of [(367, 2), 673, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2761, 734, F27, 32) (dual of [734, 673, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(29) [i] based on
- linear OA(2760, 729, F27, 32) (dual of [729, 669, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2756, 729, F27, 30) (dual of [729, 673, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(31) ⊂ Ce(29) [i] based on
- OOA 2-folding [i] based on linear OA(2761, 734, F27, 32) (dual of [734, 673, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(2761, 367, F27, 2, 32) (dual of [(367, 2), 673, 33]-NRT-code), using
(29, 29+32, 74919)-Net in Base 27 — Upper bound on s
There is no (29, 61, 74920)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 2057 173410 277799 258175 015678 331712 991031 077165 097850 047781 693623 017993 939831 220155 640833 > 2761 [i]