Best Known (17−8, 17, s)-Nets in Base 81
(17−8, 17, 1642)-Net over F81 — Constructive and digital
Digital (9, 17, 1642)-net over F81, using
- net defined by OOA [i] based on linear OOA(8117, 1642, F81, 8, 8) (dual of [(1642, 8), 13119, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8117, 6568, F81, 8) (dual of [6568, 6551, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, 6569, F81, 8) (dual of [6569, 6552, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(819, 6561, F81, 5) (dual of [6561, 6552, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(8117, 6569, F81, 8) (dual of [6569, 6552, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8117, 6568, F81, 8) (dual of [6568, 6551, 9]-code), using
(17−8, 17, 4594)-Net over F81 — Digital
Digital (9, 17, 4594)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8117, 4594, F81, 8) (dual of [4594, 4577, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, 6569, F81, 8) (dual of [6569, 6552, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(819, 6561, F81, 5) (dual of [6561, 6552, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(8117, 6569, F81, 8) (dual of [6569, 6552, 9]-code), using
(17−8, 17, 3572925)-Net in Base 81 — Upper bound on s
There is no (9, 17, 3572926)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 278 128533 459560 965457 708317 262721 > 8117 [i]