Best Known (54−18, 54, s)-Nets in Base 27
(54−18, 54, 2188)-Net over F27 — Constructive and digital
Digital (36, 54, 2188)-net over F27, using
- net defined by OOA [i] based on linear OOA(2754, 2188, F27, 18, 18) (dual of [(2188, 18), 39330, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2754, 19692, F27, 18) (dual of [19692, 19638, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2754, 19694, F27, 18) (dual of [19694, 19640, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2743, 19683, F27, 15) (dual of [19683, 19640, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2754, 19694, F27, 18) (dual of [19694, 19640, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2754, 19692, F27, 18) (dual of [19692, 19638, 19]-code), using
(54−18, 54, 14412)-Net over F27 — Digital
Digital (36, 54, 14412)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2754, 14412, F27, 18) (dual of [14412, 14358, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2754, 19694, F27, 18) (dual of [19694, 19640, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2743, 19683, F27, 15) (dual of [19683, 19640, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2754, 19694, F27, 18) (dual of [19694, 19640, 19]-code), using
(54−18, 54, large)-Net in Base 27 — Upper bound on s
There is no (36, 54, large)-net in base 27, because
- 16 times m-reduction [i] would yield (36, 38, large)-net in base 27, but