Best Known (78−9, 78, s)-Nets in Base 3
(78−9, 78, 398580)-Net over F3 — Constructive and digital
Digital (69, 78, 398580)-net over F3, using
- net defined by OOA [i] based on linear OOA(378, 398580, F3, 9, 9) (dual of [(398580, 9), 3587142, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(378, 1594321, F3, 9) (dual of [1594321, 1594243, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(378, 1594322, F3, 9) (dual of [1594322, 1594244, 10]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(378, 1594322, F3, 9) (dual of [1594322, 1594244, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(378, 1594321, F3, 9) (dual of [1594321, 1594243, 10]-code), using
(78−9, 78, 797161)-Net over F3 — Digital
Digital (69, 78, 797161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(378, 797161, F3, 2, 9) (dual of [(797161, 2), 1594244, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(378, 1594322, F3, 9) (dual of [1594322, 1594244, 10]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using
- OOA 2-folding [i] based on linear OA(378, 1594322, F3, 9) (dual of [1594322, 1594244, 10]-code), using
(78−9, 78, large)-Net in Base 3 — Upper bound on s
There is no (69, 78, large)-net in base 3, because
- 7 times m-reduction [i] would yield (69, 71, large)-net in base 3, but