Best Known (167, 189, s)-Nets in Base 3
(167, 189, 144941)-Net over F3 — Constructive and digital
Digital (167, 189, 144941)-net over F3, using
- 32 times duplication [i] based on digital (165, 187, 144941)-net over F3, using
- net defined by OOA [i] based on linear OOA(3187, 144941, F3, 22, 22) (dual of [(144941, 22), 3188515, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3187, 1594351, F3, 22) (dual of [1594351, 1594164, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, 1594353, F3, 22) (dual of [1594353, 1594166, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3187, 1594353, F3, 22) (dual of [1594353, 1594166, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3187, 1594351, F3, 22) (dual of [1594351, 1594164, 23]-code), using
- net defined by OOA [i] based on linear OOA(3187, 144941, F3, 22, 22) (dual of [(144941, 22), 3188515, 23]-NRT-code), using
(167, 189, 398588)-Net over F3 — Digital
Digital (167, 189, 398588)-net over F3, using
- 32 times duplication [i] based on digital (165, 187, 398588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 398588, F3, 4, 22) (dual of [(398588, 4), 1594165, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3187, 1594352, F3, 22) (dual of [1594352, 1594165, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, 1594353, F3, 22) (dual of [1594353, 1594166, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3187, 1594353, F3, 22) (dual of [1594353, 1594166, 23]-code), using
- OOA 4-folding [i] based on linear OA(3187, 1594352, F3, 22) (dual of [1594352, 1594165, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 398588, F3, 4, 22) (dual of [(398588, 4), 1594165, 23]-NRT-code), using
(167, 189, large)-Net in Base 3 — Upper bound on s
There is no (167, 189, large)-net in base 3, because
- 20 times m-reduction [i] would yield (167, 169, large)-net in base 3, but