Best Known (133, 133+17, s)-Nets in Base 3
(133, 133+17, 199295)-Net over F3 — Constructive and digital
Digital (133, 150, 199295)-net over F3, using
- net defined by OOA [i] based on linear OOA(3150, 199295, F3, 17, 17) (dual of [(199295, 17), 3387865, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3150, 1594361, F3, 17) (dual of [1594361, 1594211, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3150, 1594368, F3, 17) (dual of [1594368, 1594218, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3150, 1594368, F3, 17) (dual of [1594368, 1594218, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3150, 1594361, F3, 17) (dual of [1594361, 1594211, 18]-code), using
(133, 133+17, 531456)-Net over F3 — Digital
Digital (133, 150, 531456)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3150, 531456, F3, 3, 17) (dual of [(531456, 3), 1594218, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3150, 1594368, F3, 17) (dual of [1594368, 1594218, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(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(16) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(3150, 1594368, F3, 17) (dual of [1594368, 1594218, 18]-code), using
(133, 133+17, large)-Net in Base 3 — Upper bound on s
There is no (133, 150, large)-net in base 3, because
- 15 times m-reduction [i] would yield (133, 135, large)-net in base 3, but