Best Known (187−28, 187, s)-Nets in Base 3
(187−28, 187, 4219)-Net over F3 — Constructive and digital
Digital (159, 187, 4219)-net over F3, using
- 32 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
(187−28, 187, 19691)-Net over F3 — Digital
Digital (159, 187, 19691)-net over F3, using
- 32 times duplication [i] based on digital (157, 185, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3185, 19691, F3, 3, 28) (dual of [(19691, 3), 58888, 29]-NRT-code), using
(187−28, 187, 7135159)-Net in Base 3 — Upper bound on s
There is no (159, 187, 7135160)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 166600 046599 147804 415697 324648 486536 464789 395089 710364 314786 815561 534991 571573 510533 564881 > 3187 [i]