Best Known (86−16, 86, s)-Nets in Base 9
(86−16, 86, 66431)-Net over F9 — Constructive and digital
Digital (70, 86, 66431)-net over F9, using
- net defined by OOA [i] based on linear OOA(986, 66431, F9, 16, 16) (dual of [(66431, 16), 1062810, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(986, 531448, F9, 16) (dual of [531448, 531362, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 531454, F9, 16) (dual of [531454, 531368, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(973, 531441, F9, 14) (dual of [531441, 531368, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 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(986, 531454, F9, 16) (dual of [531454, 531368, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(986, 531448, F9, 16) (dual of [531448, 531362, 17]-code), using
(86−16, 86, 469868)-Net over F9 — Digital
Digital (70, 86, 469868)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(986, 469868, F9, 16) (dual of [469868, 469782, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 531454, F9, 16) (dual of [531454, 531368, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(973, 531441, F9, 14) (dual of [531441, 531368, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 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(986, 531454, F9, 16) (dual of [531454, 531368, 17]-code), using
(86−16, 86, large)-Net in Base 9 — Upper bound on s
There is no (70, 86, large)-net in base 9, because
- 14 times m-reduction [i] would yield (70, 72, large)-net in base 9, but