Best Known (71, 87, s)-Nets in Base 3
(71, 87, 823)-Net over F3 — Constructive and digital
Digital (71, 87, 823)-net over F3, using
- net defined by OOA [i] based on linear OOA(387, 823, F3, 16, 16) (dual of [(823, 16), 13081, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(387, 6584, F3, 16) (dual of [6584, 6497, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [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(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(34, 21, F3, 2) (dual of [21, 17, 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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- OA 8-folding and stacking [i] based on linear OA(387, 6584, F3, 16) (dual of [6584, 6497, 17]-code), using
(71, 87, 3292)-Net over F3 — Digital
Digital (71, 87, 3292)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(387, 3292, F3, 2, 16) (dual of [(3292, 2), 6497, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(387, 6584, F3, 16) (dual of [6584, 6497, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [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(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(34, 21, F3, 2) (dual of [21, 17, 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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(387, 6584, F3, 16) (dual of [6584, 6497, 17]-code), using
(71, 87, 290631)-Net in Base 3 — Upper bound on s
There is no (71, 87, 290632)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 323263 404645 830724 628242 519345 024136 895745 > 387 [i]