Best Known (70−12, 70, s)-Nets in Base 9
(70−12, 70, 177149)-Net over F9 — Constructive and digital
Digital (58, 70, 177149)-net over F9, using
- net defined by OOA [i] based on linear OOA(970, 177149, F9, 12, 12) (dual of [(177149, 12), 2125718, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(970, 1062894, F9, 12) (dual of [1062894, 1062824, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(970, 1062896, F9, 12) (dual of [1062896, 1062826, 13]-code), using
- trace code [i] based on linear OA(8135, 531448, F81, 12) (dual of [531448, 531413, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8135, 531448, F81, 12) (dual of [531448, 531413, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(970, 1062896, F9, 12) (dual of [1062896, 1062826, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(970, 1062894, F9, 12) (dual of [1062894, 1062824, 13]-code), using
(70−12, 70, 1062896)-Net over F9 — Digital
Digital (58, 70, 1062896)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(970, 1062896, F9, 12) (dual of [1062896, 1062826, 13]-code), using
- trace code [i] based on linear OA(8135, 531448, F81, 12) (dual of [531448, 531413, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8135, 531448, F81, 12) (dual of [531448, 531413, 13]-code), using
(70−12, 70, large)-Net in Base 9 — Upper bound on s
There is no (58, 70, large)-net in base 9, because
- 10 times m-reduction [i] would yield (58, 60, large)-net in base 9, but