Best Known (68−8, 68, s)-Nets in Base 3
(68−8, 68, 398584)-Net over F3 — Constructive and digital
Digital (60, 68, 398584)-net over F3, using
- 32 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(366, 398584, F3, 8, 8) (dual of [(398584, 8), 3188606, 9]-NRT-code), using
(68−8, 68, 797169)-Net over F3 — Digital
Digital (60, 68, 797169)-net over F3, using
- 31 times duplication [i] based on digital (59, 67, 797169)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(367, 797169, F3, 2, 8) (dual of [(797169, 2), 1594271, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(367, 1594338, F3, 8) (dual of [1594338, 1594271, 9]-code), using
- construction X4 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(314, 15, F3, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
- dual of repetition code with length 15 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(367, 1594338, F3, 8) (dual of [1594338, 1594271, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(367, 797169, F3, 2, 8) (dual of [(797169, 2), 1594271, 9]-NRT-code), using
(68−8, 68, large)-Net in Base 3 — Upper bound on s
There is no (60, 68, large)-net in base 3, because
- 6 times m-reduction [i] would yield (60, 62, large)-net in base 3, but