Best Known (12, 12+12, s)-Nets in Base 27
(12, 12+12, 122)-Net over F27 — Constructive and digital
Digital (12, 24, 122)-net over F27, using
- 271 times duplication [i] based on digital (11, 23, 122)-net over F27, using
- net defined by OOA [i] based on linear OOA(2723, 122, F27, 12, 12) (dual of [(122, 12), 1441, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2723, 732, F27, 12) (dual of [732, 709, 13]-code), using
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- OA 6-folding and stacking [i] based on linear OA(2723, 732, F27, 12) (dual of [732, 709, 13]-code), using
- net defined by OOA [i] based on linear OOA(2723, 122, F27, 12, 12) (dual of [(122, 12), 1441, 13]-NRT-code), using
(12, 12+12, 164)-Net in Base 27 — Constructive
(12, 24, 164)-net in base 27, using
- base change [i] based on digital (6, 18, 164)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (0, 12, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (0, 6, 82)-net over F81, using
- (u, u+v)-construction [i] based on
(12, 12+12, 367)-Net over F27 — Digital
Digital (12, 24, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2724, 367, F27, 2, 12) (dual of [(367, 2), 710, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2724, 734, F27, 12) (dual of [734, 710, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(2723, 729, F27, 12) (dual of [729, 706, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2719, 729, F27, 10) (dual of [729, 710, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(2724, 734, F27, 12) (dual of [734, 710, 13]-code), using
(12, 12+12, 61190)-Net in Base 27 — Upper bound on s
There is no (12, 24, 61191)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 22529 376797 873393 887969 843861 962805 > 2724 [i]