Best Known (164−22, 164, s)-Nets in Base 3
(164−22, 164, 16108)-Net over F3 — Constructive and digital
Digital (142, 164, 16108)-net over F3, using
- 31 times duplication [i] based on digital (141, 163, 16108)-net over F3, using
- net defined by OOA [i] based on linear OOA(3163, 16108, F3, 22, 22) (dual of [(16108, 22), 354213, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3163, 177188, F3, 22) (dual of [177188, 177025, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, 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
- OA 11-folding and stacking [i] based on linear OA(3163, 177188, F3, 22) (dual of [177188, 177025, 23]-code), using
- net defined by OOA [i] based on linear OOA(3163, 16108, F3, 22, 22) (dual of [(16108, 22), 354213, 23]-NRT-code), using
(164−22, 164, 59063)-Net over F3 — Digital
Digital (142, 164, 59063)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 59063, F3, 3, 22) (dual of [(59063, 3), 177025, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3164, 177189, F3, 22) (dual of [177189, 177025, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3163, 177188, F3, 22) (dual of [177188, 177025, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, 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
- 1 times code embedding in larger space [i] based on linear OA(3163, 177188, F3, 22) (dual of [177188, 177025, 23]-code), using
- OOA 3-folding [i] based on linear OA(3164, 177189, F3, 22) (dual of [177189, 177025, 23]-code), using
(164−22, 164, large)-Net in Base 3 — Upper bound on s
There is no (142, 164, large)-net in base 3, because
- 20 times m-reduction [i] would yield (142, 144, large)-net in base 3, but