Best Known (89−16, 89, s)-Nets in Base 5
(89−16, 89, 9768)-Net over F5 — Constructive and digital
Digital (73, 89, 9768)-net over F5, using
- net defined by OOA [i] based on linear OOA(589, 9768, F5, 16, 16) (dual of [(9768, 16), 156199, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(589, 78144, F5, 16) (dual of [78144, 78055, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(589, 78150, F5, 16) (dual of [78150, 78061, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [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(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(589, 78150, F5, 16) (dual of [78150, 78061, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(589, 78144, F5, 16) (dual of [78144, 78055, 17]-code), using
(89−16, 89, 39075)-Net over F5 — Digital
Digital (73, 89, 39075)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(589, 39075, F5, 2, 16) (dual of [(39075, 2), 78061, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(589, 78150, F5, 16) (dual of [78150, 78061, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [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(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 25, F5, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(589, 78150, F5, 16) (dual of [78150, 78061, 17]-code), using
(89−16, 89, large)-Net in Base 5 — Upper bound on s
There is no (73, 89, large)-net in base 5, because
- 14 times m-reduction [i] would yield (73, 75, large)-net in base 5, but