Best Known (56−11, 56, s)-Nets in Base 9
(56−11, 56, 106289)-Net over F9 — Constructive and digital
Digital (45, 56, 106289)-net over F9, using
- 91 times duplication [i] based on digital (44, 55, 106289)-net over F9, using
- net defined by OOA [i] based on linear OOA(955, 106289, F9, 11, 11) (dual of [(106289, 11), 1169124, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(955, 531446, F9, 11) (dual of [531446, 531391, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(955, 531447, F9, 11) (dual of [531447, 531392, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(955, 531447, F9, 11) (dual of [531447, 531392, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(955, 531446, F9, 11) (dual of [531446, 531391, 12]-code), using
- net defined by OOA [i] based on linear OOA(955, 106289, F9, 11, 11) (dual of [(106289, 11), 1169124, 12]-NRT-code), using
(56−11, 56, 351672)-Net over F9 — Digital
Digital (45, 56, 351672)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(956, 351672, F9, 11) (dual of [351672, 351616, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(956, 531448, F9, 11) (dual of [531448, 531392, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(955, 531447, F9, 11) (dual of [531447, 531392, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 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(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(10) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(955, 531447, F9, 11) (dual of [531447, 531392, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(956, 531448, F9, 11) (dual of [531448, 531392, 12]-code), using
(56−11, 56, large)-Net in Base 9 — Upper bound on s
There is no (45, 56, large)-net in base 9, because
- 9 times m-reduction [i] would yield (45, 47, large)-net in base 9, but