Best Known (9, 9+10, s)-Nets in Base 27
(9, 9+10, 146)-Net over F27 — Constructive and digital
Digital (9, 19, 146)-net over F27, using
- net defined by OOA [i] based on linear OOA(2719, 146, F27, 10, 10) (dual of [(146, 10), 1441, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2719, 730, F27, 10) (dual of [730, 711, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2719, 731, F27, 10) (dual of [731, 712, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [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(2717, 729, F27, 9) (dual of [729, 712, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,8], 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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(2719, 731, F27, 10) (dual of [731, 712, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2719, 730, F27, 10) (dual of [730, 711, 11]-code), using
(9, 9+10, 150)-Net in Base 27 — Constructive
(9, 19, 150)-net in base 27, using
- 1 times m-reduction [i] based on (9, 20, 150)-net in base 27, using
- base change [i] based on digital (4, 15, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- base change [i] based on digital (4, 15, 150)-net over F81, using
(9, 9+10, 366)-Net over F27 — Digital
Digital (9, 19, 366)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2719, 366, F27, 2, 10) (dual of [(366, 2), 713, 11]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
(9, 9+10, 27542)-Net in Base 27 — Upper bound on s
There is no (9, 19, 27543)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1570 053797 331559 324956 400815 > 2719 [i]