Best Known (108−16, 108, s)-Nets in Base 3
(108−16, 108, 7384)-Net over F3 — Constructive and digital
Digital (92, 108, 7384)-net over F3, using
- 33 times duplication [i] based on digital (89, 105, 7384)-net over F3, using
- net defined by OOA [i] based on linear OOA(3105, 7384, F3, 16, 16) (dual of [(7384, 16), 118039, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3105, 59072, F3, 16) (dual of [59072, 58967, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3105, 59073, F3, 16) (dual of [59073, 58968, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3105, 59073, F3, 16) (dual of [59073, 58968, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3105, 59072, F3, 16) (dual of [59072, 58967, 17]-code), using
- net defined by OOA [i] based on linear OOA(3105, 7384, F3, 16, 16) (dual of [(7384, 16), 118039, 17]-NRT-code), using
(108−16, 108, 22005)-Net over F3 — Digital
Digital (92, 108, 22005)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3108, 22005, F3, 2, 16) (dual of [(22005, 2), 43902, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3108, 29538, F3, 2, 16) (dual of [(29538, 2), 58968, 17]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3106, 29537, F3, 2, 16) (dual of [(29537, 2), 58968, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 59074, F3, 16) (dual of [59074, 58968, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3105, 59073, F3, 16) (dual of [59073, 58968, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(15) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3105, 59073, F3, 16) (dual of [59073, 58968, 17]-code), using
- OOA 2-folding [i] based on linear OA(3106, 59074, F3, 16) (dual of [59074, 58968, 17]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3106, 29537, F3, 2, 16) (dual of [(29537, 2), 58968, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3108, 29538, F3, 2, 16) (dual of [(29538, 2), 58968, 17]-NRT-code), using
(108−16, 108, 5197522)-Net in Base 3 — Upper bound on s
There is no (92, 108, 5197523)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3381 394283 186482 916849 257754 878421 693426 879081 523441 > 3108 [i]