Best Known (86, 86+12, s)-Nets in Base 3
(86, 86+12, 88577)-Net over F3 — Constructive and digital
Digital (86, 98, 88577)-net over F3, using
- net defined by OOA [i] based on linear OOA(398, 88577, F3, 12, 12) (dual of [(88577, 12), 1062826, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(398, 531462, F3, 12) (dual of [531462, 531364, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(398, 531466, F3, 12) (dual of [531466, 531368, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(31, 25, F3, 1) (dual of [25, 24, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(398, 531466, F3, 12) (dual of [531466, 531368, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(398, 531462, F3, 12) (dual of [531462, 531364, 13]-code), using
(86, 86+12, 254684)-Net over F3 — Digital
Digital (86, 98, 254684)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 254684, F3, 2, 12) (dual of [(254684, 2), 509270, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(398, 265733, F3, 2, 12) (dual of [(265733, 2), 531368, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(398, 531466, F3, 12) (dual of [531466, 531368, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(31, 25, F3, 1) (dual of [25, 24, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(398, 531466, F3, 12) (dual of [531466, 531368, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(398, 265733, F3, 2, 12) (dual of [(265733, 2), 531368, 13]-NRT-code), using
(86, 86+12, large)-Net in Base 3 — Upper bound on s
There is no (86, 98, large)-net in base 3, because
- 10 times m-reduction [i] would yield (86, 88, large)-net in base 3, but