Best Known (10−4, 10, s)-Nets in Base 81
(10−4, 10, 265722)-Net over F81 — Constructive and digital
Digital (6, 10, 265722)-net over F81, using
- net defined by OOA [i] based on linear OOA(8110, 265722, F81, 4, 4) (dual of [(265722, 4), 1062878, 5]-NRT-code), using
- OA 2-folding and stacking [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
- OA 2-folding and stacking [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
(10−4, 10, 531444)-Net over F81 — Digital
Digital (6, 10, 531444)-net over F81, using
- net defined by OOA [i] based on linear OOA(8110, 531444, F81, 4, 4) (dual of [(531444, 4), 2125766, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8110, 531444, F81, 3, 4) (dual of [(531444, 3), 1594322, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8110, 531444, F81, 4) (dual of [531444, 531434, 5]-code), using
- appending kth column [i] based on linear OOA(8110, 531444, F81, 3, 4) (dual of [(531444, 3), 1594322, 5]-NRT-code), using
(10−4, 10, large)-Net in Base 81 — Upper bound on s
There is no (6, 10, large)-net in base 81, because
- 2 times m-reduction [i] would yield (6, 8, large)-net in base 81, but