Best Known (55−19, 55, s)-Nets in Base 27
(55−19, 55, 2187)-Net over F27 — Constructive and digital
Digital (36, 55, 2187)-net over F27, using
- net defined by OOA [i] based on linear OOA(2755, 2187, F27, 19, 19) (dual of [(2187, 19), 41498, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using
(55−19, 55, 9843)-Net over F27 — Digital
Digital (36, 55, 9843)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2755, 9843, F27, 2, 19) (dual of [(9843, 2), 19631, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2755, 19686, F27, 19) (dual of [19686, 19631, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(2755, 19683, F27, 19) (dual of [19683, 19628, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(270, 3, F27, 0) (dual of [3, 3, 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(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(2755, 19686, F27, 19) (dual of [19686, 19631, 20]-code), using
(55−19, 55, large)-Net in Base 27 — Upper bound on s
There is no (36, 55, large)-net in base 27, because
- 17 times m-reduction [i] would yield (36, 38, large)-net in base 27, but