Best Known (19−9, 19, s)-Nets in Base 27
(19−9, 19, 184)-Net over F27 — Constructive and digital
Digital (10, 19, 184)-net over F27, using
- net defined by OOA [i] based on linear OOA(2719, 184, F27, 9, 9) (dual of [(184, 9), 1637, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2719, 737, F27, 9) (dual of [737, 718, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- 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(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(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(2719, 737, F27, 9) (dual of [737, 718, 10]-code), using
(19−9, 19, 620)-Net over F27 — Digital
Digital (10, 19, 620)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2719, 620, F27, 9) (dual of [620, 601, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2719, 737, F27, 9) (dual of [737, 718, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- 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(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(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(2719, 737, F27, 9) (dual of [737, 718, 10]-code), using
(19−9, 19, 235078)-Net in Base 27 — Upper bound on s
There is no (10, 19, 235079)-net in base 27, because
- 1 times m-reduction [i] would yield (10, 18, 235079)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 58 149998 793581 267245 845017 > 2718 [i]