Best Known (104, 104+18, s)-Nets in Base 3
(104, 104+18, 6563)-Net over F3 — Constructive and digital
Digital (104, 122, 6563)-net over F3, using
- net defined by OOA [i] based on linear OOA(3122, 6563, F3, 18, 18) (dual of [(6563, 18), 118012, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3122, 59067, F3, 18) (dual of [59067, 58945, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 59070, F3, 18) (dual of [59070, 58948, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(31, 21, F3, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3122, 59070, F3, 18) (dual of [59070, 58948, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3122, 59067, F3, 18) (dual of [59067, 58945, 19]-code), using
(104, 104+18, 21058)-Net over F3 — Digital
Digital (104, 122, 21058)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 21058, F3, 2, 18) (dual of [(21058, 2), 41994, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3122, 29535, F3, 2, 18) (dual of [(29535, 2), 58948, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3122, 59070, F3, 18) (dual of [59070, 58948, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(31, 21, F3, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3122, 59070, F3, 18) (dual of [59070, 58948, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(3122, 29535, F3, 2, 18) (dual of [(29535, 2), 58948, 19]-NRT-code), using
(104, 104+18, 6086456)-Net in Base 3 — Upper bound on s
There is no (104, 122, 6086457)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 16173 105140 263631 839660 960294 088139 004754 136577 365991 111635 > 3122 [i]