Best Known (97−16, 97, s)-Nets in Base 3
(97−16, 97, 2463)-Net over F3 — Constructive and digital
Digital (81, 97, 2463)-net over F3, using
- 32 times duplication [i] based on digital (79, 95, 2463)-net over F3, using
- net defined by OOA [i] based on linear OOA(395, 2463, F3, 16, 16) (dual of [(2463, 16), 39313, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(395, 19704, F3, 16) (dual of [19704, 19609, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(395, 19705, F3, 16) (dual of [19705, 19610, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(395, 19705, F3, 16) (dual of [19705, 19610, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(395, 19704, F3, 16) (dual of [19704, 19609, 17]-code), using
- net defined by OOA [i] based on linear OOA(395, 2463, F3, 16, 16) (dual of [(2463, 16), 39313, 17]-NRT-code), using
(97−16, 97, 8678)-Net over F3 — Digital
Digital (81, 97, 8678)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(397, 8678, F3, 2, 16) (dual of [(8678, 2), 17259, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(397, 9853, F3, 2, 16) (dual of [(9853, 2), 19609, 17]-NRT-code), using
- 31 times duplication [i] based on linear OOA(396, 9853, F3, 2, 16) (dual of [(9853, 2), 19610, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(396, 19706, F3, 16) (dual of [19706, 19610, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(395, 19705, F3, 16) (dual of [19705, 19610, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(395, 19705, F3, 16) (dual of [19705, 19610, 17]-code), using
- OOA 2-folding [i] based on linear OA(396, 19706, F3, 16) (dual of [19706, 19610, 17]-code), using
- 31 times duplication [i] based on linear OOA(396, 9853, F3, 2, 16) (dual of [(9853, 2), 19610, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(397, 9853, F3, 2, 16) (dual of [(9853, 2), 19609, 17]-NRT-code), using
(97−16, 97, 1147499)-Net in Base 3 — Upper bound on s
There is no (81, 97, 1147500)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 19088 144043 338444 795768 913670 892051 043271 704001 > 397 [i]