Best Known (54, 54+27, s)-Nets in Base 27
(54, 54+27, 1514)-Net over F27 — Constructive and digital
Digital (54, 81, 1514)-net over F27, using
- 272 times duplication [i] based on digital (52, 79, 1514)-net over F27, using
- net defined by OOA [i] based on linear OOA(2779, 1514, F27, 27, 27) (dual of [(1514, 27), 40799, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2779, 19683, F27, 27) (dual of [19683, 19604, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2779, 19683, F27, 27) (dual of [19683, 19604, 28]-code), using
- net defined by OOA [i] based on linear OOA(2779, 1514, F27, 27, 27) (dual of [(1514, 27), 40799, 28]-NRT-code), using
(54, 54+27, 14882)-Net over F27 — Digital
Digital (54, 81, 14882)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2781, 14882, F27, 27) (dual of [14882, 14801, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, 19694, F27, 27) (dual of [19694, 19613, 28]-code), using
- 1 times truncation [i] based on linear OA(2782, 19695, F27, 28) (dual of [19695, 19613, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(273, 12, F27, 3) (dual of [12, 9, 4]-code or 12-arc in PG(2,27) or 12-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(27) ⊂ Ce(23) [i] based on
- 1 times truncation [i] based on linear OA(2782, 19695, F27, 28) (dual of [19695, 19613, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, 19694, F27, 27) (dual of [19694, 19613, 28]-code), using
(54, 54+27, large)-Net in Base 27 — Upper bound on s
There is no (54, 81, large)-net in base 27, because
- 25 times m-reduction [i] would yield (54, 56, large)-net in base 27, but