Best Known (74−6, 74, s)-Nets in Base 3
(74−6, 74, 3188656)-Net over F3 — Constructive and digital
Digital (68, 74, 3188656)-net over F3, using
- trace code for nets [i] based on digital (31, 37, 1594328)-net over F9, using
- net defined by OOA [i] based on linear OOA(937, 1594328, F9, 6, 6) (dual of [(1594328, 6), 9565931, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(937, 4782984, F9, 6) (dual of [4782984, 4782947, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [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(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(937, 4782984, F9, 6) (dual of [4782984, 4782947, 7]-code), using
- net defined by OOA [i] based on linear OOA(937, 1594328, F9, 6, 6) (dual of [(1594328, 6), 9565931, 7]-NRT-code), using
(74−6, 74, large)-Net over F3 — Digital
Digital (68, 74, large)-net over F3, using
- 31 times duplication [i] based on digital (67, 73, large)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (66, 73, large)-net over F3, using
(74−6, 74, large)-Net in Base 3 — Upper bound on s
There is no (68, 74, large)-net in base 3, because
- 4 times m-reduction [i] would yield (68, 70, large)-net in base 3, but