Best Known (26, 26+19, s)-Nets in Base 81
(26, 26+19, 731)-Net over F81 — Constructive and digital
Digital (26, 45, 731)-net over F81, using
- 812 times duplication [i] based on digital (24, 43, 731)-net over F81, using
- net defined by OOA [i] based on linear OOA(8143, 731, F81, 19, 19) (dual of [(731, 19), 13846, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8143, 6580, F81, 19) (dual of [6580, 6537, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8143, 6581, F81, 19) (dual of [6581, 6538, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- linear OA(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(8143, 6581, F81, 19) (dual of [6581, 6538, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8143, 6580, F81, 19) (dual of [6580, 6537, 20]-code), using
- net defined by OOA [i] based on linear OOA(8143, 731, F81, 19, 19) (dual of [(731, 19), 13846, 20]-NRT-code), using
(26, 26+19, 6587)-Net over F81 — Digital
Digital (26, 45, 6587)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8145, 6587, F81, 19) (dual of [6587, 6542, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(9) [i] based on
- linear OA(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(818, 26, F81, 8) (dual of [26, 18, 9]-code or 26-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(18) ⊂ Ce(9) [i] based on
(26, 26+19, large)-Net in Base 81 — Upper bound on s
There is no (26, 45, large)-net in base 81, because
- 17 times m-reduction [i] would yield (26, 28, large)-net in base 81, but