Best Known (28, 28+20, s)-Nets in Base 81
(28, 28+20, 659)-Net over F81 — Constructive and digital
Digital (28, 48, 659)-net over F81, using
- net defined by OOA [i] based on linear OOA(8148, 659, F81, 20, 20) (dual of [(659, 20), 13132, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8148, 6590, F81, 20) (dual of [6590, 6542, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(9) [i] based on
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to Ce(19) ⊂ Ce(9) [i] based on
- OA 10-folding and stacking [i] based on linear OA(8148, 6590, F81, 20) (dual of [6590, 6542, 21]-code), using
(28, 28+20, 6590)-Net over F81 — Digital
Digital (28, 48, 6590)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8148, 6590, F81, 20) (dual of [6590, 6542, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(9) [i] based on
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to Ce(19) ⊂ Ce(9) [i] based on
(28, 28+20, large)-Net in Base 81 — Upper bound on s
There is no (28, 48, large)-net in base 81, because
- 18 times m-reduction [i] would yield (28, 30, large)-net in base 81, but