Best Known (21−4, 21, s)-Nets in Base 9
(21−4, 21, 531444)-Net over F9 — Constructive and digital
Digital (17, 21, 531444)-net over F9, using
- 91 times duplication [i] based on digital (16, 20, 531444)-net over F9, using
- net defined by OOA [i] based on linear OOA(920, 531444, F9, 4, 4) (dual of [(531444, 4), 2125756, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(920, 531444, F9, 3, 4) (dual of [(531444, 3), 1594312, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(920, 1062888, F9, 4) (dual of [1062888, 1062868, 5]-code), using
- trace code [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(817, 531441, F81, 3) (dual of [531441, 531434, 4]-code or 531441-cap in PG(6,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(920, 1062888, F9, 4) (dual of [1062888, 1062868, 5]-code), using
- appending kth column [i] based on linear OOA(920, 531444, F9, 3, 4) (dual of [(531444, 3), 1594312, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(920, 531444, F9, 4, 4) (dual of [(531444, 4), 2125756, 5]-NRT-code), using
(21−4, 21, 1086405)-Net over F9 — Digital
Digital (17, 21, 1086405)-net over F9, using
- net defined by OOA [i] based on linear OOA(921, 1086405, F9, 4, 4) (dual of [(1086405, 4), 4345599, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(921, 1086405, F9, 3, 4) (dual of [(1086405, 3), 3259194, 5]-NRT-code), using
(21−4, 21, large)-Net in Base 9 — Upper bound on s
There is no (17, 21, large)-net in base 9, because
- 2 times m-reduction [i] would yield (17, 19, large)-net in base 9, but