Best Known (111−12, 111, s)-Nets in Base 3
(111−12, 111, 265728)-Net over F3 — Constructive and digital
Digital (99, 111, 265728)-net over F3, using
- net defined by OOA [i] based on linear OOA(3111, 265728, F3, 12, 12) (dual of [(265728, 12), 3188625, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3111, 1594368, F3, 12) (dual of [1594368, 1594257, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(12) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(3111, 1594368, F3, 12) (dual of [1594368, 1594257, 13]-code), using
(111−12, 111, 797184)-Net over F3 — Digital
Digital (99, 111, 797184)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3111, 797184, F3, 2, 12) (dual of [(797184, 2), 1594257, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3111, 1594368, F3, 12) (dual of [1594368, 1594257, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(12) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(3111, 1594368, F3, 12) (dual of [1594368, 1594257, 13]-code), using
(111−12, 111, large)-Net in Base 3 — Upper bound on s
There is no (99, 111, large)-net in base 3, because
- 10 times m-reduction [i] would yield (99, 101, large)-net in base 3, but