Best Known (177, 177+16, s)-Nets in Base 3
(177, 177+16, 1050217)-Net over F3 — Constructive and digital
Digital (177, 193, 1050217)-net over F3, using
- 31 times duplication [i] based on digital (176, 192, 1050217)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (33, 41, 1642)-net over F3, using
- net defined by OOA [i] based on linear OOA(341, 1642, F3, 8, 8) (dual of [(1642, 8), 13095, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(341, 6568, F3, 8) (dual of [6568, 6527, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 6569, F3, 8) (dual of [6569, 6528, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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
- discarding factors / shortening the dual code based on linear OA(341, 6569, F3, 8) (dual of [6569, 6528, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(341, 6568, F3, 8) (dual of [6568, 6527, 9]-code), using
- net defined by OOA [i] based on linear OOA(341, 1642, F3, 8, 8) (dual of [(1642, 8), 13095, 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 (33, 41, 1642)-net over F3, using
- (u, u+v)-construction [i] based on
(177, 177+16, large)-Net over F3 — Digital
Digital (177, 193, large)-net over F3, using
- 32 times duplication [i] based on digital (175, 191, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3191, large, F3, 16) (dual of [large, large−191, 17]-code), using
- 40 times code embedding in larger space [i] 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]
- 40 times code embedding in larger space [i] based on linear OA(3151, large, F3, 16) (dual of [large, large−151, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3191, large, F3, 16) (dual of [large, large−191, 17]-code), using
(177, 177+16, large)-Net in Base 3 — Upper bound on s
There is no (177, 193, large)-net in base 3, because
- 14 times m-reduction [i] would yield (177, 179, large)-net in base 3, but