Best Known (138, 138+22, s)-Nets in Base 3
(138, 138+22, 16106)-Net over F3 — Constructive and digital
Digital (138, 160, 16106)-net over F3, using
- 31 times duplication [i] based on digital (137, 159, 16106)-net over F3, using
- net defined by OOA [i] based on linear OOA(3159, 16106, F3, 22, 22) (dual of [(16106, 22), 354173, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3159, 177166, F3, 22) (dual of [177166, 177007, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 177173, F3, 22) (dual of [177173, 177014, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3159, 177173, F3, 22) (dual of [177173, 177014, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3159, 177166, F3, 22) (dual of [177166, 177007, 23]-code), using
- net defined by OOA [i] based on linear OOA(3159, 16106, F3, 22, 22) (dual of [(16106, 22), 354173, 23]-NRT-code), using
(138, 138+22, 54767)-Net over F3 — Digital
Digital (138, 160, 54767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3160, 54767, F3, 3, 22) (dual of [(54767, 3), 164141, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3160, 59058, F3, 3, 22) (dual of [(59058, 3), 177014, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3160, 177174, F3, 22) (dual of [177174, 177014, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3159, 177173, F3, 22) (dual of [177173, 177014, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3133, 177147, F3, 19) (dual of [177147, 177014, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(21) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3159, 177173, F3, 22) (dual of [177173, 177014, 23]-code), using
- OOA 3-folding [i] based on linear OA(3160, 177174, F3, 22) (dual of [177174, 177014, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(3160, 59058, F3, 3, 22) (dual of [(59058, 3), 177014, 23]-NRT-code), using
(138, 138+22, large)-Net in Base 3 — Upper bound on s
There is no (138, 160, large)-net in base 3, because
- 20 times m-reduction [i] would yield (138, 140, large)-net in base 3, but