Best Known (12, 12+10, s)-Nets in Base 27
(12, 12+10, 148)-Net over F27 — Constructive and digital
Digital (12, 22, 148)-net over F27, using
- net defined by OOA [i] based on linear OOA(2722, 148, F27, 10, 10) (dual of [(148, 10), 1458, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2722, 740, F27, 10) (dual of [740, 718, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- 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(2711, 729, F27, 6) (dual of [729, 718, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(273, 11, F27, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- OA 5-folding and stacking [i] based on linear OA(2722, 740, F27, 10) (dual of [740, 718, 11]-code), using
(12, 12+10, 200)-Net in Base 27 — Constructive
(12, 22, 200)-net in base 27, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- (5, 15, 100)-net in base 27, using
- 1 times m-reduction [i] based on (5, 16, 100)-net in base 27, using
- base change [i] based on digital (1, 12, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- base change [i] based on digital (1, 12, 100)-net over F81, using
- 1 times m-reduction [i] based on (5, 16, 100)-net in base 27, using
- digital (2, 7, 351)-net over F27, using
(12, 12+10, 755)-Net over F27 — Digital
Digital (12, 22, 755)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2722, 755, F27, 10) (dual of [755, 733, 11]-code), using
- 20 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 16 times 0) [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
- construction XX applied to C1 = C([727,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([727,8]) [i] based on
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [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([727,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([727,8]) [i] based on
- 20 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 16 times 0) [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
(12, 12+10, 199002)-Net in Base 27 — Upper bound on s
There is no (12, 22, 199003)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 30 903490 359749 554509 653090 391447 > 2722 [i]