Best Known (216−24, 216, s)-Nets in Base 3
(216−24, 216, 132864)-Net over F3 — Constructive and digital
Digital (192, 216, 132864)-net over F3, using
- 31 times duplication [i] based on digital (191, 215, 132864)-net over F3, using
- net defined by OOA [i] based on linear OOA(3215, 132864, F3, 24, 24) (dual of [(132864, 24), 3188521, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3215, 1594368, F3, 24) (dual of [1594368, 1594153, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- OA 12-folding and stacking [i] based on linear OA(3215, 1594368, F3, 24) (dual of [1594368, 1594153, 25]-code), using
- net defined by OOA [i] based on linear OOA(3215, 132864, F3, 24, 24) (dual of [(132864, 24), 3188521, 25]-NRT-code), using
(216−24, 216, 500727)-Net over F3 — Digital
Digital (192, 216, 500727)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3216, 500727, F3, 3, 24) (dual of [(500727, 3), 1501965, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3216, 531456, F3, 3, 24) (dual of [(531456, 3), 1594152, 25]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3215, 531456, F3, 3, 24) (dual of [(531456, 3), 1594153, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3215, 1594368, F3, 24) (dual of [1594368, 1594153, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(3215, 1594368, F3, 24) (dual of [1594368, 1594153, 25]-code), using
- 31 times duplication [i] based on linear OOA(3215, 531456, F3, 3, 24) (dual of [(531456, 3), 1594153, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3216, 531456, F3, 3, 24) (dual of [(531456, 3), 1594152, 25]-NRT-code), using
(216−24, 216, large)-Net in Base 3 — Upper bound on s
There is no (192, 216, large)-net in base 3, because
- 22 times m-reduction [i] would yield (192, 194, large)-net in base 3, but