Best Known (221−16, 221, s)-Nets in Base 3
(221−16, 221, 1447163)-Net over F3 — Constructive and digital
Digital (205, 221, 1447163)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (62, 70, 398588)-net over F3, using
- net defined by OOA [i] based on linear OOA(370, 398588, F3, 8, 8) (dual of [(398588, 8), 3188634, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(370, 1594352, F3, 8) (dual of [1594352, 1594282, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(370, 1594353, F3, 8) (dual of [1594353, 1594283, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [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(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(370, 1594353, F3, 8) (dual of [1594353, 1594283, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(370, 1594352, F3, 8) (dual of [1594352, 1594282, 9]-code), using
- net defined by OOA [i] based on linear OOA(370, 398588, F3, 8, 8) (dual of [(398588, 8), 3188634, 9]-NRT-code), using
- digital (135, 151, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3151, 8388600, F3, 16) (dual of [8388600, 8388449, 17]-code), using
- net defined by OOA [i] based on linear OOA(3151, 1048575, F3, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
- digital (62, 70, 398588)-net over F3, using
(221−16, 221, large)-Net over F3 — Digital
Digital (205, 221, large)-net over F3, using
- 35 times duplication [i] based on digital (200, 216, large)-net over F3, using
- t-expansion [i] based on digital (198, 216, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- t-expansion [i] based on digital (198, 216, large)-net over F3, using
(221−16, 221, large)-Net in Base 3 — Upper bound on s
There is no (205, 221, large)-net in base 3, because
- 14 times m-reduction [i] would yield (205, 207, large)-net in base 3, but