Best Known (66−8, 66, s)-Nets in Base 3
(66−8, 66, 398584)-Net over F3 — Constructive and digital
Digital (58, 66, 398584)-net over F3, using
- net defined by OOA [i] based on linear OOA(366, 398584, F3, 8, 8) (dual of [(398584, 8), 3188606, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(366, 1594336, F3, 8) (dual of [1594336, 1594270, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- OA 4-folding and stacking [i] based on linear OA(366, 1594336, F3, 8) (dual of [1594336, 1594270, 9]-code), using
(66−8, 66, 797168)-Net over F3 — Digital
Digital (58, 66, 797168)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(366, 797168, F3, 2, 8) (dual of [(797168, 2), 1594270, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(366, 1594336, F3, 8) (dual of [1594336, 1594270, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(366, 1594336, F3, 8) (dual of [1594336, 1594270, 9]-code), using
(66−8, 66, large)-Net in Base 3 — Upper bound on s
There is no (58, 66, large)-net in base 3, because
- 6 times m-reduction [i] would yield (58, 60, large)-net in base 3, but