Best Known (20, 39, s)-Nets in Base 81
(20, 39, 729)-Net over F81 — Constructive and digital
Digital (20, 39, 729)-net over F81, using
- 812 times duplication [i] based on digital (18, 37, 729)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 729, F81, 19, 19) (dual of [(729, 19), 13814, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using
- net defined by OOA [i] based on linear OOA(8137, 729, F81, 19, 19) (dual of [(729, 19), 13814, 20]-NRT-code), using
(20, 39, 2195)-Net over F81 — Digital
Digital (20, 39, 2195)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8139, 2195, F81, 2, 19) (dual of [(2195, 2), 4351, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8139, 3284, F81, 2, 19) (dual of [(3284, 2), 6529, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8139, 6568, F81, 19) (dual of [6568, 6529, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 6569, F81, 19) (dual of [6569, 6530, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [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(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(8139, 6569, F81, 19) (dual of [6569, 6530, 20]-code), using
- OOA 2-folding [i] based on linear OA(8139, 6568, F81, 19) (dual of [6568, 6529, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(8139, 3284, F81, 2, 19) (dual of [(3284, 2), 6529, 20]-NRT-code), using
(20, 39, 5925281)-Net in Base 81 — Upper bound on s
There is no (20, 39, 5925282)-net in base 81, because
- 1 times m-reduction [i] would yield (20, 38, 5925282)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 3 329896 865040 272067 017187 976466 207660 448997 302724 654142 553479 774806 926241 > 8138 [i]