Best Known (57, 57+12, s)-Nets in Base 9
(57, 57+12, 177148)-Net over F9 — Constructive and digital
Digital (57, 69, 177148)-net over F9, using
- 91 times duplication [i] based on digital (56, 68, 177148)-net over F9, using
- net defined by OOA [i] based on linear OOA(968, 177148, F9, 12, 12) (dual of [(177148, 12), 2125708, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(968, 1062888, F9, 12) (dual of [1062888, 1062820, 13]-code), using
- trace code [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [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(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(968, 1062888, F9, 12) (dual of [1062888, 1062820, 13]-code), using
- net defined by OOA [i] based on linear OOA(968, 177148, F9, 12, 12) (dual of [(177148, 12), 2125708, 13]-NRT-code), using
(57, 57+12, 1062890)-Net over F9 — Digital
Digital (57, 69, 1062890)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(969, 1062890, F9, 12) (dual of [1062890, 1062821, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(968, 1062888, F9, 12) (dual of [1062888, 1062820, 13]-code), using
- trace code [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [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(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- trace code [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- linear OA(968, 1062889, F9, 11) (dual of [1062889, 1062821, 12]-code), using Gilbert–Varšamov bound and bm = 968 > Vbs−1(k−1) = 544 495130 045382 750141 091579 608363 210520 123825 113576 369566 751297 [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(968, 1062888, F9, 12) (dual of [1062888, 1062820, 13]-code), using
- construction X with Varšamov bound [i] based on
(57, 57+12, large)-Net in Base 9 — Upper bound on s
There is no (57, 69, large)-net in base 9, because
- 10 times m-reduction [i] would yield (57, 59, large)-net in base 9, but