Best Known (52, 52+22, s)-Nets in Base 81
(52, 52+22, 48316)-Net over F81 — Constructive and digital
Digital (52, 74, 48316)-net over F81, using
- 812 times duplication [i] based on digital (50, 72, 48316)-net over F81, using
- net defined by OOA [i] based on linear OOA(8172, 48316, F81, 22, 22) (dual of [(48316, 22), 1062880, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8172, 531476, F81, 22) (dual of [531476, 531404, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(12) [i] based on
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8137, 531441, F81, 13) (dual of [531441, 531404, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(818, 35, F81, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,81)), using
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- Reed–Solomon code RS(73,81) [i]
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- construction X applied to Ce(21) ⊂ Ce(12) [i] based on
- OA 11-folding and stacking [i] based on linear OA(8172, 531476, F81, 22) (dual of [531476, 531404, 23]-code), using
- net defined by OOA [i] based on linear OOA(8172, 48316, F81, 22, 22) (dual of [(48316, 22), 1062880, 23]-NRT-code), using
(52, 52+22, 576167)-Net over F81 — Digital
Digital (52, 74, 576167)-net over F81, using
(52, 52+22, large)-Net in Base 81 — Upper bound on s
There is no (52, 74, large)-net in base 81, because
- 20 times m-reduction [i] would yield (52, 54, large)-net in base 81, but