Best Known (66, 79, s)-Nets in Base 3
(66, 79, 3284)-Net over F3 — Constructive and digital
Digital (66, 79, 3284)-net over F3, using
- 32 times duplication [i] based on digital (64, 77, 3284)-net over F3, using
- net defined by OOA [i] based on linear OOA(377, 3284, F3, 13, 13) (dual of [(3284, 13), 42615, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(377, 19705, F3, 13) (dual of [19705, 19628, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(377, 19705, F3, 13) (dual of [19705, 19628, 14]-code), using
- net defined by OOA [i] based on linear OOA(377, 3284, F3, 13, 13) (dual of [(3284, 13), 42615, 14]-NRT-code), using
(66, 79, 9854)-Net over F3 — Digital
Digital (66, 79, 9854)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(379, 9854, F3, 2, 13) (dual of [(9854, 2), 19629, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(379, 19708, F3, 13) (dual of [19708, 19629, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- 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(355, 19684, F3, 9) (dual of [19684, 19629, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 24, F3, 3) (dual of [24, 18, 4]-code or 24-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 C([0,6]) ⊂ C([0,4]) [i] based on
- OOA 2-folding [i] based on linear OA(379, 19708, F3, 13) (dual of [19708, 19629, 14]-code), using
(66, 79, 2386532)-Net in Base 3 — Upper bound on s
There is no (66, 79, 2386533)-net in base 3, because
- 1 times m-reduction [i] would yield (66, 78, 2386533)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16 423213 803425 215015 688148 316347 390465 > 378 [i]