Best Known (181−27, 181, s)-Nets in Base 3
(181−27, 181, 4542)-Net over F3 — Constructive and digital
Digital (154, 181, 4542)-net over F3, using
- 31 times duplication [i] based on digital (153, 180, 4542)-net over F3, using
- net defined by OOA [i] based on linear OOA(3180, 4542, F3, 27, 27) (dual of [(4542, 27), 122454, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3180, 59047, F3, 27) (dual of [59047, 58867, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3180, 59048, F3, 27) (dual of [59048, 58868, 28]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3180, 59048, F3, 27) (dual of [59048, 58868, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3180, 59047, F3, 27) (dual of [59047, 58867, 28]-code), using
- net defined by OOA [i] based on linear OOA(3180, 4542, F3, 27, 27) (dual of [(4542, 27), 122454, 28]-NRT-code), using
(181−27, 181, 19686)-Net over F3 — Digital
Digital (154, 181, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 19686, F3, 3, 27) (dual of [(19686, 3), 58877, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3181, 59058, F3, 27) (dual of [59058, 58877, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 59059, F3, 27) (dual of [59059, 58878, 28]-code), using
- 1 times truncation [i] based on linear OA(3182, 59060, F3, 28) (dual of [59060, 58878, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [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(3171, 59049, F3, 26) (dual of [59049, 58878, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- 1 times truncation [i] based on linear OA(3182, 59060, F3, 28) (dual of [59060, 58878, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3181, 59059, F3, 27) (dual of [59059, 58878, 28]-code), using
- OOA 3-folding [i] based on linear OA(3181, 59058, F3, 27) (dual of [59058, 58877, 28]-code), using
(181−27, 181, large)-Net in Base 3 — Upper bound on s
There is no (154, 181, large)-net in base 3, because
- 25 times m-reduction [i] would yield (154, 156, large)-net in base 3, but