Best Known (116−18, 116, s)-Nets in Base 3
(116−18, 116, 2190)-Net over F3 — Constructive and digital
Digital (98, 116, 2190)-net over F3, using
- 1 times m-reduction [i] based on digital (98, 117, 2190)-net over F3, using
- net defined by OOA [i] based on linear OOA(3117, 2190, F3, 19, 19) (dual of [(2190, 19), 41493, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3117, 19711, F3, 19) (dual of [19711, 19594, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3117, 19712, F3, 19) (dual of [19712, 19595, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(373, 19684, F3, 13) (dual of [19684, 19611, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3117, 19712, F3, 19) (dual of [19712, 19595, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3117, 19711, F3, 19) (dual of [19711, 19594, 20]-code), using
- net defined by OOA [i] based on linear OOA(3117, 2190, F3, 19, 19) (dual of [(2190, 19), 41493, 20]-NRT-code), using
(116−18, 116, 9858)-Net over F3 — Digital
Digital (98, 116, 9858)-net over F3, using
- 31 times duplication [i] based on digital (97, 115, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3115, 9858, F3, 2, 18) (dual of [(9858, 2), 19601, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3115, 19716, F3, 18) (dual of [19716, 19601, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3115, 19716, F3, 18) (dual of [19716, 19601, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3115, 9858, F3, 2, 18) (dual of [(9858, 2), 19601, 19]-NRT-code), using
(116−18, 116, 2926058)-Net in Base 3 — Upper bound on s
There is no (98, 116, 2926059)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 22 185335 049450 174010 875832 241557 749658 851852 896610 537095 > 3116 [i]