Best Known (57, 57+29, s)-Nets in Base 27
(57, 57+29, 1407)-Net over F27 — Constructive and digital
Digital (57, 86, 1407)-net over F27, using
- net defined by OOA [i] based on linear OOA(2786, 1407, F27, 29, 29) (dual of [(1407, 29), 40717, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2786, 19699, F27, 29) (dual of [19699, 19613, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(2782, 19683, F27, 29) (dual of [19683, 19601, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(274, 16, F27, 4) (dual of [16, 12, 5]-code or 16-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(2786, 19699, F27, 29) (dual of [19699, 19613, 30]-code), using
(57, 57+29, 13463)-Net over F27 — Digital
Digital (57, 86, 13463)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2786, 13463, F27, 29) (dual of [13463, 13377, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2786, 19691, F27, 29) (dual of [19691, 19605, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- linear OA(2785, 19684, F27, 29) (dual of [19684, 19599, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- 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]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,14]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2786, 19691, F27, 29) (dual of [19691, 19605, 30]-code), using
(57, 57+29, large)-Net in Base 27 — Upper bound on s
There is no (57, 86, large)-net in base 27, because
- 27 times m-reduction [i] would yield (57, 59, large)-net in base 27, but