Best Known (171, 171+22, s)-Nets in Base 3
(171, 171+22, 144942)-Net over F3 — Constructive and digital
Digital (171, 193, 144942)-net over F3, using
- 32 times duplication [i] based on digital (169, 191, 144942)-net over F3, using
- net defined by OOA [i] based on linear OOA(3191, 144942, F3, 22, 22) (dual of [(144942, 22), 3188533, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3191, 1594362, F3, 22) (dual of [1594362, 1594171, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 1594364, F3, 22) (dual of [1594364, 1594173, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [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(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(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3191, 1594364, F3, 22) (dual of [1594364, 1594173, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3191, 1594362, F3, 22) (dual of [1594362, 1594171, 23]-code), using
- net defined by OOA [i] based on linear OOA(3191, 144942, F3, 22, 22) (dual of [(144942, 22), 3188533, 23]-NRT-code), using
(171, 171+22, 410546)-Net over F3 — Digital
Digital (171, 193, 410546)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 410546, F3, 3, 22) (dual of [(410546, 3), 1231445, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 531457, F3, 3, 22) (dual of [(531457, 3), 1594178, 23]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3192, 531457, F3, 3, 22) (dual of [(531457, 3), 1594179, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3192, 1594371, F3, 22) (dual of [1594371, 1594179, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [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(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(39, 48, F3, 4) (dual of [48, 39, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3192, 1594371, F3, 22) (dual of [1594371, 1594179, 23]-code), using
- 31 times duplication [i] based on linear OOA(3192, 531457, F3, 3, 22) (dual of [(531457, 3), 1594179, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3193, 531457, F3, 3, 22) (dual of [(531457, 3), 1594178, 23]-NRT-code), using
(171, 171+22, large)-Net in Base 3 — Upper bound on s
There is no (171, 193, large)-net in base 3, because
- 20 times m-reduction [i] would yield (171, 173, large)-net in base 3, but