Best Known (165, 165+20, s)-Nets in Base 3
(165, 165+20, 478298)-Net over F3 — Constructive and digital
Digital (165, 185, 478298)-net over F3, using
- 32 times duplication [i] based on digital (163, 183, 478298)-net over F3, using
- net defined by OOA [i] based on linear OOA(3183, 478298, F3, 20, 20) (dual of [(478298, 20), 9565777, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3183, 4782980, F3, 20) (dual of [4782980, 4782797, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3183, 4782983, F3, 20) (dual of [4782983, 4782800, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3183, 4782983, F3, 20) (dual of [4782983, 4782800, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3183, 4782980, F3, 20) (dual of [4782980, 4782797, 21]-code), using
- net defined by OOA [i] based on linear OOA(3183, 478298, F3, 20, 20) (dual of [(478298, 20), 9565777, 21]-NRT-code), using
(165, 165+20, 1195746)-Net over F3 — Digital
Digital (165, 185, 1195746)-net over F3, using
- 31 times duplication [i] based on digital (164, 184, 1195746)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3184, 1195746, F3, 4, 20) (dual of [(1195746, 4), 4782800, 21]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3184, 4782984, F3, 20) (dual of [4782984, 4782800, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3183, 4782983, F3, 20) (dual of [4782983, 4782800, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 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(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3183, 4782983, F3, 20) (dual of [4782983, 4782800, 21]-code), using
- OOA 4-folding [i] based on linear OA(3184, 4782984, F3, 20) (dual of [4782984, 4782800, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3184, 1195746, F3, 4, 20) (dual of [(1195746, 4), 4782800, 21]-NRT-code), using
(165, 165+20, large)-Net in Base 3 — Upper bound on s
There is no (165, 185, large)-net in base 3, because
- 18 times m-reduction [i] would yield (165, 167, large)-net in base 3, but