Best Known (67, 67+15, s)-Nets in Base 9
(67, 67+15, 75923)-Net over F9 — Constructive and digital
Digital (67, 82, 75923)-net over F9, using
- net defined by OOA [i] based on linear OOA(982, 75923, F9, 15, 15) (dual of [(75923, 15), 1138763, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(982, 531462, F9, 15) (dual of [531462, 531380, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(979, 531441, F9, 15) (dual of [531441, 531362, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(982, 531462, F9, 15) (dual of [531462, 531380, 16]-code), using
(67, 67+15, 531462)-Net over F9 — Digital
Digital (67, 82, 531462)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(982, 531462, F9, 15) (dual of [531462, 531380, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(979, 531441, F9, 15) (dual of [531441, 531362, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(14) ⊂ Ce(11) [i] based on
(67, 67+15, large)-Net in Base 9 — Upper bound on s
There is no (67, 82, large)-net in base 9, because
- 13 times m-reduction [i] would yield (67, 69, large)-net in base 9, but