Best Known (40−14, 40, s)-Nets in Base 81
(40−14, 40, 75920)-Net over F81 — Constructive and digital
Digital (26, 40, 75920)-net over F81, using
- net defined by OOA [i] based on linear OOA(8140, 75920, F81, 14, 14) (dual of [(75920, 14), 1062840, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8140, 531440, F81, 14) (dual of [531440, 531400, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8140, 531440, F81, 14) (dual of [531440, 531400, 15]-code), using
(40−14, 40, 240361)-Net over F81 — Digital
Digital (26, 40, 240361)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8140, 240361, F81, 2, 14) (dual of [(240361, 2), 480682, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8140, 265722, F81, 2, 14) (dual of [(265722, 2), 531404, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8140, 531444, F81, 14) (dual of [531444, 531404, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(8137, 531441, F81, 13) (dual of [531441, 531404, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(8140, 531444, F81, 14) (dual of [531444, 531404, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(8140, 265722, F81, 2, 14) (dual of [(265722, 2), 531404, 15]-NRT-code), using
(40−14, 40, large)-Net in Base 81 — Upper bound on s
There is no (26, 40, large)-net in base 81, because
- 12 times m-reduction [i] would yield (26, 28, large)-net in base 81, but