Best Known (78−12, 78, s)-Nets in Base 3
(78−12, 78, 3284)-Net over F3 — Constructive and digital
Digital (66, 78, 3284)-net over F3, using
- 31 times duplication [i] based on digital (65, 77, 3284)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (64, 77, 3284)-net over F3, using
(78−12, 78, 10676)-Net over F3 — Digital
Digital (66, 78, 10676)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(378, 10676, F3, 12) (dual of [10676, 10598, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(378, 19709, F3, 12) (dual of [19709, 19631, 13]-code), using
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [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(346, 19683, F3, 8) (dual of [19683, 19637, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(378, 19709, F3, 12) (dual of [19709, 19631, 13]-code), using
(78−12, 78, 2386532)-Net in Base 3 — Upper bound on s
There is no (66, 78, 2386533)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 16 423213 803425 215015 688148 316347 390465 > 378 [i]