Best Known (15, 15+4, s)-Nets in Base 9
(15, 15+4, 265723)-Net over F9 — Constructive and digital
Digital (15, 19, 265723)-net over F9, using
- net defined by OOA [i] based on linear OOA(919, 265723, F9, 4, 4) (dual of [(265723, 4), 1062873, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(919, 265723, F9, 3, 4) (dual of [(265723, 3), 797150, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(919, 531446, F9, 4) (dual of [531446, 531427, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(919, 531447, F9, 4) (dual of [531447, 531428, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(913, 531441, F9, 3) (dual of [531441, 531428, 4]-code or 531441-cap in PG(12,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(919, 531447, F9, 4) (dual of [531447, 531428, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(919, 531446, F9, 4) (dual of [531446, 531427, 5]-code), using
- appending kth column [i] based on linear OOA(919, 265723, F9, 3, 4) (dual of [(265723, 3), 797150, 5]-NRT-code), using
(15, 15+4, 531447)-Net over F9 — Digital
Digital (15, 19, 531447)-net over F9, using
- net defined by OOA [i] based on linear OOA(919, 531447, F9, 4, 4) (dual of [(531447, 4), 2125769, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(919, 531447, F9, 3, 4) (dual of [(531447, 3), 1594322, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(919, 531447, F9, 4) (dual of [531447, 531428, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(913, 531441, F9, 3) (dual of [531441, 531428, 4]-code or 531441-cap in PG(12,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(919, 531447, F9, 4) (dual of [531447, 531428, 5]-code), using
- appending kth column [i] based on linear OOA(919, 531447, F9, 3, 4) (dual of [(531447, 3), 1594322, 5]-NRT-code), using
(15, 15+4, large)-Net in Base 9 — Upper bound on s
There is no (15, 19, large)-net in base 9, because
- 2 times m-reduction [i] would yield (15, 17, large)-net in base 9, but