Best Known (19, 37, s)-Nets in Base 81
(19, 37, 729)-Net over F81 — Constructive and digital
Digital (19, 37, 729)-net over F81, using
- t-expansion [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
(19, 37, 2273)-Net over F81 — Digital
Digital (19, 37, 2273)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8137, 2273, F81, 2, 18) (dual of [(2273, 2), 4509, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8137, 3284, F81, 2, 18) (dual of [(3284, 2), 6531, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8137, 6568, F81, 18) (dual of [6568, 6531, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, 6569, F81, 18) (dual of [6569, 6532, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8137, 6569, F81, 18) (dual of [6569, 6532, 19]-code), using
- OOA 2-folding [i] based on linear OA(8137, 6568, F81, 18) (dual of [6568, 6531, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(8137, 3284, F81, 2, 18) (dual of [(3284, 2), 6531, 19]-NRT-code), using
(19, 37, 3636260)-Net in Base 81 — Upper bound on s
There is no (19, 37, 3636261)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 41109 926747 576562 281652 254435 061125 219554 717230 580321 508915 735860 962321 > 8137 [i]