Best Known (12, 12+8, s)-Nets in Base 27
(12, 12+8, 756)-Net over F27 — Constructive and digital
Digital (12, 20, 756)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 28)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 8, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 0, 28)-net over F27, using
(12, 12+8, 1610)-Net over F27 — Digital
Digital (12, 20, 1610)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2720, 1610, F27, 8) (dual of [1610, 1590, 9]-code), using
- 873 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0, 1, 64 times 0, 1, 241 times 0, 1, 553 times 0) [i] based on linear OA(2715, 732, F27, 8) (dual of [732, 717, 9]-code), using
- construction XX applied to C1 = C([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- 873 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0, 1, 64 times 0, 1, 241 times 0, 1, 553 times 0) [i] based on linear OA(2715, 732, F27, 8) (dual of [732, 717, 9]-code), using
(12, 12+8, 1640)-Net in Base 27 — Constructive
(12, 20, 1640)-net in base 27, using
- base change [i] based on digital (7, 15, 1640)-net over F81, using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
(12, 12+8, 2982)-Net in Base 27
(12, 20, 2982)-net in base 27, using
- base change [i] based on digital (7, 15, 2982)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8115, 2982, F81, 2, 8) (dual of [(2982, 2), 5949, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8115, 3281, F81, 2, 8) (dual of [(3281, 2), 6547, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8115, 6562, F81, 8) (dual of [6562, 6547, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- OOA 2-folding [i] based on linear OA(8115, 6562, F81, 8) (dual of [6562, 6547, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(8115, 3281, F81, 2, 8) (dual of [(3281, 2), 6547, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8115, 2982, F81, 2, 8) (dual of [(2982, 2), 5949, 9]-NRT-code), using
(12, 12+8, 1221511)-Net in Base 27 — Upper bound on s
There is no (12, 20, 1221512)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 42391 174886 371921 090622 536513 > 2720 [i]