Best Known (66, 66+6, s)-Nets in Base 3
(66, 66+6, 3188650)-Net over F3 — Constructive and digital
Digital (66, 72, 3188650)-net over F3, using
- trace code for nets [i] based on digital (30, 36, 1594325)-net over F9, using
- net defined by OOA [i] based on linear OOA(936, 1594325, F9, 6, 6) (dual of [(1594325, 6), 9565914, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(936, 4782975, F9, 6) (dual of [4782975, 4782939, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(936, 4782976, F9, 6) (dual of [4782976, 4782940, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(936, 4782969, F9, 6) (dual of [4782969, 4782933, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(929, 4782969, F9, 5) (dual of [4782969, 4782940, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(936, 4782976, F9, 6) (dual of [4782976, 4782940, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(936, 4782975, F9, 6) (dual of [4782975, 4782939, 7]-code), using
- net defined by OOA [i] based on linear OOA(936, 1594325, F9, 6, 6) (dual of [(1594325, 6), 9565914, 7]-NRT-code), using
(66, 66+6, large)-Net over F3 — Digital
Digital (66, 72, large)-net over F3, using
- 1 times m-reduction [i] based on digital (66, 73, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(373, large, F3, 7) (dual of [large, large−73, 8]-code), using
- 12 times code embedding in larger space [i] based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 12 times code embedding in larger space [i] based on linear OA(361, large, F3, 7) (dual of [large, large−61, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(373, large, F3, 7) (dual of [large, large−73, 8]-code), using
(66, 66+6, large)-Net in Base 3 — Upper bound on s
There is no (66, 72, large)-net in base 3, because
- 4 times m-reduction [i] would yield (66, 68, large)-net in base 3, but