Best Known (25−5, 25, s)-Nets in Base 9
(25−5, 25, 265723)-Net over F9 — Constructive and digital
Digital (20, 25, 265723)-net over F9, using
- net defined by OOA [i] based on linear OOA(925, 265723, F9, 5, 5) (dual of [(265723, 5), 1328590, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(925, 531447, F9, 5) (dual of [531447, 531422, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(925, 531441, F9, 5) (dual of [531441, 531416, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(919, 531441, F9, 4) (dual of [531441, 531422, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(90, 6, F9, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(925, 531447, F9, 5) (dual of [531447, 531422, 6]-code), using
(25−5, 25, 531447)-Net over F9 — Digital
Digital (20, 25, 531447)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(925, 531447, F9, 5) (dual of [531447, 531422, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(925, 531441, F9, 5) (dual of [531441, 531416, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(919, 531441, F9, 4) (dual of [531441, 531422, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(90, 6, F9, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
(25−5, 25, large)-Net in Base 9 — Upper bound on s
There is no (20, 25, large)-net in base 9, because
- 3 times m-reduction [i] would yield (20, 22, large)-net in base 9, but