Best Known (88−16, 88, s)-Nets in Base 3
(88−16, 88, 823)-Net over F3 — Constructive and digital
Digital (72, 88, 823)-net over F3, using
- 31 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(387, 823, F3, 16, 16) (dual of [(823, 16), 13081, 17]-NRT-code), using
(88−16, 88, 3294)-Net over F3 — Digital
Digital (72, 88, 3294)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(388, 3294, F3, 2, 16) (dual of [(3294, 2), 6500, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(388, 6588, F3, 16) (dual of [6588, 6500, 17]-code), using
- construction X applied to Ce(15) ⊂ 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(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(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(388, 6588, F3, 16) (dual of [6588, 6500, 17]-code), using
(88−16, 88, 333414)-Net in Base 3 — Upper bound on s
There is no (72, 88, 333415)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 969792 160935 162671 153783 554431 603446 351217 > 388 [i]