Best Known (111, 111+22, s)-Nets in Base 3
(111, 111+22, 1791)-Net over F3 — Constructive and digital
Digital (111, 133, 1791)-net over F3, using
- 32 times duplication [i] based on digital (109, 131, 1791)-net over F3, using
- net defined by OOA [i] based on linear OOA(3131, 1791, F3, 22, 22) (dual of [(1791, 22), 39271, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3131, 19701, F3, 22) (dual of [19701, 19570, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3131, 19705, F3, 22) (dual of [19705, 19574, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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(3131, 19705, F3, 22) (dual of [19705, 19574, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3131, 19701, F3, 22) (dual of [19701, 19570, 23]-code), using
- net defined by OOA [i] based on linear OOA(3131, 1791, F3, 22, 22) (dual of [(1791, 22), 39271, 23]-NRT-code), using
(111, 111+22, 7705)-Net over F3 — Digital
Digital (111, 133, 7705)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3133, 7705, F3, 2, 22) (dual of [(7705, 2), 15277, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3133, 9853, F3, 2, 22) (dual of [(9853, 2), 19573, 23]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3132, 9853, F3, 2, 22) (dual of [(9853, 2), 19574, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3132, 19706, F3, 22) (dual of [19706, 19574, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3131, 19705, F3, 22) (dual of [19705, 19574, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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(3131, 19705, F3, 22) (dual of [19705, 19574, 23]-code), using
- OOA 2-folding [i] based on linear OA(3132, 19706, F3, 22) (dual of [19706, 19574, 23]-code), using
- 31 times duplication [i] based on linear OOA(3132, 9853, F3, 2, 22) (dual of [(9853, 2), 19574, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3133, 9853, F3, 2, 22) (dual of [(9853, 2), 19573, 23]-NRT-code), using
(111, 111+22, 1441486)-Net in Base 3 — Upper bound on s
There is no (111, 133, 1441487)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2865 016074 268273 556804 412946 373961 154253 001186 192644 217736 078635 > 3133 [i]