Best Known (64, 64+33, s)-Nets in Base 27
(64, 64+33, 1231)-Net over F27 — Constructive and digital
Digital (64, 97, 1231)-net over F27, using
- net defined by OOA [i] based on linear OOA(2797, 1231, F27, 33, 33) (dual of [(1231, 33), 40526, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2797, 19697, F27, 33) (dual of [19697, 19600, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2797, 19698, F27, 33) (dual of [19698, 19601, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- linear OA(2794, 19683, F27, 33) (dual of [19683, 19589, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(273, 15, F27, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,27) or 15-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(32) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2797, 19698, F27, 33) (dual of [19698, 19601, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2797, 19697, F27, 33) (dual of [19697, 19600, 34]-code), using
(64, 64+33, 12917)-Net over F27 — Digital
Digital (64, 97, 12917)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2797, 12917, F27, 33) (dual of [12917, 12820, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2797, 19684, F27, 33) (dual of [19684, 19587, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2797, 19684, F27, 33) (dual of [19684, 19587, 34]-code), using
(64, 64+33, large)-Net in Base 27 — Upper bound on s
There is no (64, 97, large)-net in base 27, because
- 31 times m-reduction [i] would yield (64, 66, large)-net in base 27, but