Best Known (68, 84, s)-Nets in Base 3
(68, 84, 821)-Net over F3 — Constructive and digital
Digital (68, 84, 821)-net over F3, using
- 32 times duplication [i] based on digital (66, 82, 821)-net over F3, using
- net defined by OOA [i] based on linear OOA(382, 821, F3, 16, 16) (dual of [(821, 16), 13054, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(382, 6568, F3, 16) (dual of [6568, 6486, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(382, 6568, F3, 16) (dual of [6568, 6486, 17]-code), using
- net defined by OOA [i] based on linear OOA(382, 821, F3, 16, 16) (dual of [(821, 16), 13054, 17]-NRT-code), using
(68, 84, 2885)-Net over F3 — Digital
Digital (68, 84, 2885)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(384, 2885, F3, 2, 16) (dual of [(2885, 2), 5686, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(384, 3287, F3, 2, 16) (dual of [(3287, 2), 6490, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(384, 6574, F3, 16) (dual of [6574, 6490, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(384, 6574, F3, 16) (dual of [6574, 6490, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(384, 3287, F3, 2, 16) (dual of [(3287, 2), 6490, 17]-NRT-code), using
(68, 84, 192493)-Net in Base 3 — Upper bound on s
There is no (68, 84, 192494)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11972 707793 292538 604640 415575 357289 200465 > 384 [i]