Best Known (85, 85+13, s)-Nets in Base 3
(85, 85+13, 88575)-Net over F3 — Constructive and digital
Digital (85, 98, 88575)-net over F3, using
- net defined by OOA [i] based on linear OOA(398, 88575, F3, 13, 13) (dual of [(88575, 13), 1151377, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(398, 531451, F3, 13) (dual of [531451, 531353, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(398, 531454, F3, 13) (dual of [531454, 531356, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(398, 531454, F3, 13) (dual of [531454, 531356, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(398, 531451, F3, 13) (dual of [531451, 531353, 14]-code), using
(85, 85+13, 177151)-Net over F3 — Digital
Digital (85, 98, 177151)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 177151, F3, 3, 13) (dual of [(177151, 3), 531355, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(398, 531453, F3, 13) (dual of [531453, 531355, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(398, 531454, F3, 13) (dual of [531454, 531356, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(398, 531454, F3, 13) (dual of [531454, 531356, 14]-code), using
- OOA 3-folding [i] based on linear OA(398, 531453, F3, 13) (dual of [531453, 531355, 14]-code), using
(85, 85+13, large)-Net in Base 3 — Upper bound on s
There is no (85, 98, large)-net in base 3, because
- 11 times m-reduction [i] would yield (85, 87, large)-net in base 3, but