Best Known (165, 165+29, s)-Nets in Base 3
(165, 165+29, 4218)-Net over F3 — Constructive and digital
Digital (165, 194, 4218)-net over F3, using
- 33 times duplication [i] based on digital (162, 191, 4218)-net over F3, using
- net defined by OOA [i] based on linear OOA(3191, 4218, F3, 29, 29) (dual of [(4218, 29), 122131, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3191, 59053, F3, 29) (dual of [59053, 58862, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3191, 59059, F3, 29) (dual of [59059, 58868, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3191, 59059, F3, 29) (dual of [59059, 58868, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3191, 59053, F3, 29) (dual of [59053, 58862, 30]-code), using
- net defined by OOA [i] based on linear OOA(3191, 4218, F3, 29, 29) (dual of [(4218, 29), 122131, 30]-NRT-code), using
(165, 165+29, 19687)-Net over F3 — Digital
Digital (165, 194, 19687)-net over F3, using
- 31 times duplication [i] based on digital (164, 193, 19687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 19687, F3, 3, 29) (dual of [(19687, 3), 58868, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3193, 59061, F3, 29) (dual of [59061, 58868, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3191, 59059, F3, 29) (dual of [59059, 58868, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(30, 10, F3, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3191, 59059, F3, 29) (dual of [59059, 58868, 30]-code), using
- OOA 3-folding [i] based on linear OA(3193, 59061, F3, 29) (dual of [59061, 58868, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3193, 19687, F3, 3, 29) (dual of [(19687, 3), 58868, 30]-NRT-code), using
(165, 165+29, large)-Net in Base 3 — Upper bound on s
There is no (165, 194, large)-net in base 3, because
- 27 times m-reduction [i] would yield (165, 167, large)-net in base 3, but