Best Known (160, 160+28, s)-Nets in Base 3
(160, 160+28, 4219)-Net over F3 — Constructive and digital
Digital (160, 188, 4219)-net over F3, using
- 33 times duplication [i] based on digital (157, 185, 4219)-net over F3, using
- net defined by OOA [i] based on linear OOA(3185, 4219, F3, 28, 28) (dual of [(4219, 28), 117947, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3185, 59066, F3, 28) (dual of [59066, 58881, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3185, 59073, F3, 28) (dual of [59073, 58888, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(27) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3185, 59073, F3, 28) (dual of [59073, 58888, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3185, 59066, F3, 28) (dual of [59066, 58881, 29]-code), using
- net defined by OOA [i] based on linear OOA(3185, 4219, F3, 28, 28) (dual of [(4219, 28), 117947, 29]-NRT-code), using
(160, 160+28, 19692)-Net over F3 — Digital
Digital (160, 188, 19692)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3188, 19692, F3, 3, 28) (dual of [(19692, 3), 58888, 29]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3185, 19691, F3, 3, 28) (dual of [(19691, 3), 58888, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3185, 59073, F3, 28) (dual of [59073, 58888, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(27) ⊂ Ce(24) [i] based on
- OOA 3-folding [i] based on linear OA(3185, 59073, F3, 28) (dual of [59073, 58888, 29]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3185, 19691, F3, 3, 28) (dual of [(19691, 3), 58888, 29]-NRT-code), using
(160, 160+28, 7717627)-Net in Base 3 — Upper bound on s
There is no (160, 188, 7717628)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 499799 709220 109687 643028 451429 652995 053540 024555 960077 894117 245039 633661 336788 279487 159081 > 3188 [i]