Best Known (69, 69+10, s)-Nets in Base 3
(69, 69+10, 318864)-Net over F3 — Constructive and digital
Digital (69, 79, 318864)-net over F3, using
- net defined by OOA [i] based on linear OOA(379, 318864, F3, 10, 10) (dual of [(318864, 10), 3188561, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(379, 1594320, F3, 10) (dual of [1594320, 1594241, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(379, 1594320, F3, 10) (dual of [1594320, 1594241, 11]-code), using
(69, 69+10, 531441)-Net over F3 — Digital
Digital (69, 79, 531441)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(379, 531441, F3, 3, 10) (dual of [(531441, 3), 1594244, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- OOA 3-folding [i] based on linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using
(69, 69+10, large)-Net in Base 3 — Upper bound on s
There is no (69, 79, large)-net in base 3, because
- 8 times m-reduction [i] would yield (69, 71, large)-net in base 3, but