Best Known (86−16, 86, s)-Nets in Base 5
(86−16, 86, 9766)-Net over F5 — Constructive and digital
Digital (70, 86, 9766)-net over F5, using
- net defined by OOA [i] based on linear OOA(586, 9766, F5, 16, 16) (dual of [(9766, 16), 156170, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(586, 78128, F5, 16) (dual of [78128, 78042, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(586, 78128, F5, 16) (dual of [78128, 78042, 17]-code), using
(86−16, 86, 39066)-Net over F5 — Digital
Digital (70, 86, 39066)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(586, 39066, F5, 2, 16) (dual of [(39066, 2), 78046, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(586, 78132, F5, 16) (dual of [78132, 78046, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- OOA 2-folding [i] based on linear OA(586, 78132, F5, 16) (dual of [78132, 78046, 17]-code), using
(86−16, 86, large)-Net in Base 5 — Upper bound on s
There is no (70, 86, large)-net in base 5, because
- 14 times m-reduction [i] would yield (70, 72, large)-net in base 5, but