Best Known (209−16, 209, s)-Nets in Base 3
(209−16, 209, 1195755)-Net over F3 — Constructive and digital
Digital (193, 209, 1195755)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 11)-net over F3, using
- digital (182, 198, 1195744)-net over F3, using
- trace code for nets [i] based on digital (83, 99, 597872)-net over F9, using
- net defined by OOA [i] based on linear OOA(999, 597872, F9, 16, 16) (dual of [(597872, 16), 9565853, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(999, 4782976, F9, 16) (dual of [4782976, 4782877, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(15) ⊂ Ce(14) [i] based on
- OA 8-folding and stacking [i] based on linear OA(999, 4782976, F9, 16) (dual of [4782976, 4782877, 17]-code), using
- net defined by OOA [i] based on linear OOA(999, 597872, F9, 16, 16) (dual of [(597872, 16), 9565853, 17]-NRT-code), using
- trace code for nets [i] based on digital (83, 99, 597872)-net over F9, using
(209−16, 209, large)-Net over F3 — Digital
Digital (193, 209, large)-net over F3, using
- 36 times duplication [i] based on digital (187, 203, large)-net over F3, using
- t-expansion [i] based on digital (186, 203, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- t-expansion [i] based on digital (186, 203, large)-net over F3, using
(209−16, 209, large)-Net in Base 3 — Upper bound on s
There is no (193, 209, large)-net in base 3, because
- 14 times m-reduction [i] would yield (193, 195, large)-net in base 3, but