Best Known (78−26, 78, s)-Nets in Base 27
(78−26, 78, 1514)-Net over F27 — Constructive and digital
Digital (52, 78, 1514)-net over F27, using
- 1 times m-reduction [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
(78−26, 78, 14734)-Net over F27 — Digital
Digital (52, 78, 14734)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2778, 14734, F27, 26) (dual of [14734, 14656, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19694, F27, 26) (dual of [19694, 19616, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2778, 19694, F27, 26) (dual of [19694, 19616, 27]-code), using
(78−26, 78, large)-Net in Base 27 — Upper bound on s
There is no (52, 78, large)-net in base 27, because
- 24 times m-reduction [i] would yield (52, 54, large)-net in base 27, but