Best Known (209−28, 209, s)-Nets in Base 3
(209−28, 209, 12656)-Net over F3 — Constructive and digital
Digital (181, 209, 12656)-net over F3, using
- 32 times duplication [i] based on digital (179, 207, 12656)-net over F3, using
- net defined by OOA [i] based on linear OOA(3207, 12656, F3, 28, 28) (dual of [(12656, 28), 354161, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3207, 177184, F3, 28) (dual of [177184, 176977, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3207, 177188, F3, 28) (dual of [177188, 176981, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3207, 177188, F3, 28) (dual of [177188, 176981, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3207, 177184, F3, 28) (dual of [177184, 176977, 29]-code), using
- net defined by OOA [i] based on linear OOA(3207, 12656, F3, 28, 28) (dual of [(12656, 28), 354161, 29]-NRT-code), using
(209−28, 209, 59063)-Net over F3 — Digital
Digital (181, 209, 59063)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3209, 59063, F3, 3, 28) (dual of [(59063, 3), 176980, 29]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3208, 59063, F3, 3, 28) (dual of [(59063, 3), 176981, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3208, 177189, F3, 28) (dual of [177189, 176981, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3207, 177188, F3, 28) (dual of [177188, 176981, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3166, 177147, F3, 23) (dual of [177147, 176981, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3207, 177188, F3, 28) (dual of [177188, 176981, 29]-code), using
- OOA 3-folding [i] based on linear OA(3208, 177189, F3, 28) (dual of [177189, 176981, 29]-code), using
- 31 times duplication [i] based on linear OOA(3208, 59063, F3, 3, 28) (dual of [(59063, 3), 176981, 29]-NRT-code), using
(209−28, 209, large)-Net in Base 3 — Upper bound on s
There is no (181, 209, large)-net in base 3, because
- 26 times m-reduction [i] would yield (181, 183, large)-net in base 3, but