Best Known (119−20, 119, s)-Nets in Base 9
(119−20, 119, 106289)-Net over F9 — Constructive and digital
Digital (99, 119, 106289)-net over F9, using
- 91 times duplication [i] based on digital (98, 118, 106289)-net over F9, using
- net defined by OOA [i] based on linear OOA(9118, 106289, F9, 20, 20) (dual of [(106289, 20), 2125662, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9118, 1062890, F9, 20) (dual of [1062890, 1062772, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9118, 1062896, F9, 20) (dual of [1062896, 1062778, 21]-code), using
- trace code [i] based on linear OA(8159, 531448, F81, 20) (dual of [531448, 531389, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(8159, 531448, F81, 20) (dual of [531448, 531389, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9118, 1062896, F9, 20) (dual of [1062896, 1062778, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9118, 1062890, F9, 20) (dual of [1062890, 1062772, 21]-code), using
- net defined by OOA [i] based on linear OOA(9118, 106289, F9, 20, 20) (dual of [(106289, 20), 2125662, 21]-NRT-code), using
(119−20, 119, 1062898)-Net over F9 — Digital
Digital (99, 119, 1062898)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9119, 1062898, F9, 20) (dual of [1062898, 1062779, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9118, 1062896, F9, 20) (dual of [1062896, 1062778, 21]-code), using
- trace code [i] based on linear OA(8159, 531448, F81, 20) (dual of [531448, 531389, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(8159, 531448, F81, 20) (dual of [531448, 531389, 21]-code), using
- linear OA(9118, 1062897, F9, 19) (dual of [1062897, 1062779, 20]-code), using Gilbert–Varšamov bound and bm = 9118 > Vbs−1(k−1) = 8 434339 098611 024449 351197 706413 105983 663740 592810 667167 862048 029472 930805 218637 571389 087672 686620 719600 025985 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 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
- linear OA(9118, 1062896, F9, 20) (dual of [1062896, 1062778, 21]-code), using
- construction X with Varšamov bound [i] based on
(119−20, 119, large)-Net in Base 9 — Upper bound on s
There is no (99, 119, large)-net in base 9, because
- 18 times m-reduction [i] would yield (99, 101, large)-net in base 9, but