Best Known (104−20, 104, s)-Nets in Base 9
(104−20, 104, 53144)-Net over F9 — Constructive and digital
Digital (84, 104, 53144)-net over F9, using
- 91 times duplication [i] based on digital (83, 103, 53144)-net over F9, using
- net defined by OOA [i] based on linear OOA(9103, 53144, F9, 20, 20) (dual of [(53144, 20), 1062777, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9103, 531440, F9, 20) (dual of [531440, 531337, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9103, 531440, F9, 20) (dual of [531440, 531337, 21]-code), using
- net defined by OOA [i] based on linear OOA(9103, 53144, F9, 20, 20) (dual of [(53144, 20), 1062777, 21]-NRT-code), using
(104−20, 104, 272528)-Net over F9 — Digital
Digital (84, 104, 272528)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9104, 272528, F9, 20) (dual of [272528, 272424, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9104, 531448, F9, 20) (dual of [531448, 531344, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(9103, 531447, F9, 20) (dual of [531447, 531344, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(997, 531441, F9, 19) (dual of [531441, 531344, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(90, 6, F9, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(9103, 531447, F9, 20) (dual of [531447, 531344, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9104, 531448, F9, 20) (dual of [531448, 531344, 21]-code), using
(104−20, 104, large)-Net in Base 9 — Upper bound on s
There is no (84, 104, large)-net in base 9, because
- 18 times m-reduction [i] would yield (84, 86, large)-net in base 9, but