Best Known (76−12, 76, s)-Nets in Base 3
(76−12, 76, 3284)-Net over F3 — Constructive and digital
Digital (64, 76, 3284)-net over F3, using
- 1 times m-reduction [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
(76−12, 76, 9852)-Net over F3 — Digital
Digital (64, 76, 9852)-net over F3, using
- 31 times duplication [i] based on digital (63, 75, 9852)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(375, 9852, F3, 2, 12) (dual of [(9852, 2), 19629, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(375, 19704, F3, 12) (dual of [19704, 19629, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(374, 19703, F3, 12) (dual of [19703, 19629, 13]-code), using
- construction X4 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(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(374, 19703, F3, 12) (dual of [19703, 19629, 13]-code), using
- OOA 2-folding [i] based on linear OA(375, 19704, F3, 12) (dual of [19704, 19629, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(375, 9852, F3, 2, 12) (dual of [(9852, 2), 19629, 13]-NRT-code), using
(76−12, 76, 1654727)-Net in Base 3 — Upper bound on s
There is no (64, 76, 1654728)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 824802 356490 260061 810992 384437 313201 > 376 [i]