Best Known (43−15, 43, s)-Nets in Base 81
(43−15, 43, 75920)-Net over F81 — Constructive and digital
Digital (28, 43, 75920)-net over F81, using
- net defined by OOA [i] based on linear OOA(8143, 75920, F81, 15, 15) (dual of [(75920, 15), 1138757, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using
(43−15, 43, 219239)-Net over F81 — Digital
Digital (28, 43, 219239)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8143, 219239, F81, 2, 15) (dual of [(219239, 2), 438435, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8143, 265722, F81, 2, 15) (dual of [(265722, 2), 531401, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8143, 531444, F81, 15) (dual of [531444, 531401, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(8143, 531444, F81, 15) (dual of [531444, 531401, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(8143, 265722, F81, 2, 15) (dual of [(265722, 2), 531401, 16]-NRT-code), using
(43−15, 43, large)-Net in Base 81 — Upper bound on s
There is no (28, 43, large)-net in base 81, because
- 13 times m-reduction [i] would yield (28, 30, large)-net in base 81, but