Best Known (164, 164+22, s)-Nets in Base 3
(164, 164+22, 144939)-Net over F3 — Constructive and digital
Digital (164, 186, 144939)-net over F3, using
- 32 times duplication [i] based on digital (162, 184, 144939)-net over F3, using
- net defined by OOA [i] based on linear OOA(3184, 144939, F3, 22, 22) (dual of [(144939, 22), 3188474, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3184, 1594329, F3, 22) (dual of [1594329, 1594145, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3184, 1594337, F3, 22) (dual of [1594337, 1594153, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [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(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(31, 14, F3, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3184, 1594337, F3, 22) (dual of [1594337, 1594153, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3184, 1594329, F3, 22) (dual of [1594329, 1594145, 23]-code), using
- net defined by OOA [i] based on linear OOA(3184, 144939, F3, 22, 22) (dual of [(144939, 22), 3188474, 23]-NRT-code), using
(164, 164+22, 398584)-Net over F3 — Digital
Digital (164, 186, 398584)-net over F3, using
- 32 times duplication [i] based on digital (162, 184, 398584)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3184, 398584, F3, 4, 22) (dual of [(398584, 4), 1594152, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3184, 1594336, F3, 22) (dual of [1594336, 1594152, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3184, 1594337, F3, 22) (dual of [1594337, 1594153, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [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(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(31, 14, F3, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3184, 1594337, F3, 22) (dual of [1594337, 1594153, 23]-code), using
- OOA 4-folding [i] based on linear OA(3184, 1594336, F3, 22) (dual of [1594336, 1594152, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3184, 398584, F3, 4, 22) (dual of [(398584, 4), 1594152, 23]-NRT-code), using
(164, 164+22, large)-Net in Base 3 — Upper bound on s
There is no (164, 186, large)-net in base 3, because
- 20 times m-reduction [i] would yield (164, 166, large)-net in base 3, but