Best Known (72, 72+16, s)-Nets in Base 9
(72, 72+16, 66432)-Net over F9 — Constructive and digital
Digital (72, 88, 66432)-net over F9, using
- 91 times duplication [i] based on digital (71, 87, 66432)-net over F9, using
- net defined by OOA [i] based on linear OOA(987, 66432, F9, 16, 16) (dual of [(66432, 16), 1062825, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(987, 531456, F9, 16) (dual of [531456, 531369, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(973, 531441, F9, 14) (dual of [531441, 531368, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(91, 14, F9, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- OA 8-folding and stacking [i] based on linear OA(987, 531456, F9, 16) (dual of [531456, 531369, 17]-code), using
- net defined by OOA [i] based on linear OOA(987, 66432, F9, 16, 16) (dual of [(66432, 16), 1062825, 17]-NRT-code), using
(72, 72+16, 531462)-Net over F9 — Digital
Digital (72, 88, 531462)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(988, 531462, F9, 16) (dual of [531462, 531374, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(93, 21, F9, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
(72, 72+16, large)-Net in Base 9 — Upper bound on s
There is no (72, 88, large)-net in base 9, because
- 14 times m-reduction [i] would yield (72, 74, large)-net in base 9, but