Best Known (146, 146+18, s)-Nets in Base 3
(146, 146+18, 177152)-Net over F3 — Constructive and digital
Digital (146, 164, 177152)-net over F3, using
- 31 times duplication [i] based on digital (145, 163, 177152)-net over F3, using
- net defined by OOA [i] based on linear OOA(3163, 177152, F3, 18, 18) (dual of [(177152, 18), 3188573, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3163, 1594368, F3, 18) (dual of [1594368, 1594205, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- OA 9-folding and stacking [i] based on linear OA(3163, 1594368, F3, 18) (dual of [1594368, 1594205, 19]-code), using
- net defined by OOA [i] based on linear OOA(3163, 177152, F3, 18, 18) (dual of [(177152, 18), 3188573, 19]-NRT-code), using
(146, 146+18, 531456)-Net over F3 — Digital
Digital (146, 164, 531456)-net over F3, using
- 31 times duplication [i] based on digital (145, 163, 531456)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 531456, F3, 3, 18) (dual of [(531456, 3), 1594205, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3163, 1594368, F3, 18) (dual of [1594368, 1594205, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(3163, 1594368, F3, 18) (dual of [1594368, 1594205, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 531456, F3, 3, 18) (dual of [(531456, 3), 1594205, 19]-NRT-code), using
(146, 146+18, large)-Net in Base 3 — Upper bound on s
There is no (146, 164, large)-net in base 3, because
- 16 times m-reduction [i] would yield (146, 148, large)-net in base 3, but