Best Known (69, 69+15, s)-Nets in Base 9
(69, 69+15, 75924)-Net over F9 — Constructive and digital
Digital (69, 84, 75924)-net over F9, using
- 91 times duplication [i] based on digital (68, 83, 75924)-net over F9, using
- net defined by OOA [i] based on linear OOA(983, 75924, F9, 15, 15) (dual of [(75924, 15), 1138777, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(983, 531469, F9, 15) (dual of [531469, 531386, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [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(955, 531441, F9, 11) (dual of [531441, 531386, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,9)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(983, 531469, F9, 15) (dual of [531469, 531386, 16]-code), using
- net defined by OOA [i] based on linear OOA(983, 75924, F9, 15, 15) (dual of [(75924, 15), 1138777, 16]-NRT-code), using
(69, 69+15, 531471)-Net over F9 — Digital
Digital (69, 84, 531471)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(984, 531471, F9, 15) (dual of [531471, 531387, 16]-code), using
- construction XX applied to Ce(14) ⊂ Ce(10) ⊂ Ce(9) [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(955, 531441, F9, 11) (dual of [531441, 531386, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 29, F9, 3) (dual of [29, 25, 4]-code or 29-cap in PG(3,9)), 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(14) ⊂ Ce(10) ⊂ Ce(9) [i] based on
(69, 69+15, large)-Net in Base 9 — Upper bound on s
There is no (69, 84, large)-net in base 9, because
- 13 times m-reduction [i] would yield (69, 71, large)-net in base 9, but