Best Known (104, 125, s)-Nets in Base 9
(104, 125, 106289)-Net over F9 — Constructive and digital
Digital (104, 125, 106289)-net over F9, using
- 91 times duplication [i] based on digital (103, 124, 106289)-net over F9, using
- net defined by OOA [i] based on linear OOA(9124, 106289, F9, 21, 21) (dual of [(106289, 21), 2231945, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9124, 1062891, F9, 21) (dual of [1062891, 1062767, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9124, 1062898, F9, 21) (dual of [1062898, 1062774, 22]-code), using
- trace code [i] based on linear OA(8162, 531449, F81, 21) (dual of [531449, 531387, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- trace code [i] based on linear OA(8162, 531449, F81, 21) (dual of [531449, 531387, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9124, 1062898, F9, 21) (dual of [1062898, 1062774, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9124, 1062891, F9, 21) (dual of [1062891, 1062767, 22]-code), using
- net defined by OOA [i] based on linear OOA(9124, 106289, F9, 21, 21) (dual of [(106289, 21), 2231945, 22]-NRT-code), using
(104, 125, 1062900)-Net over F9 — Digital
Digital (104, 125, 1062900)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9125, 1062900, F9, 21) (dual of [1062900, 1062775, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9124, 1062898, F9, 21) (dual of [1062898, 1062774, 22]-code), using
- trace code [i] based on linear OA(8162, 531449, F81, 21) (dual of [531449, 531387, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- trace code [i] based on linear OA(8162, 531449, F81, 21) (dual of [531449, 531387, 22]-code), using
- linear OA(9124, 1062899, F9, 20) (dual of [1062899, 1062775, 21]-code), using Gilbert–Varšamov bound and bm = 9124 > Vbs−1(k−1) = 3 774734 751311 834091 713400 248430 496415 461021 788445 208254 160849 911962 039544 766177 288727 188732 276098 082231 731123 599825 [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(9124, 1062898, F9, 21) (dual of [1062898, 1062774, 22]-code), using
- construction X with Varšamov bound [i] based on
(104, 125, large)-Net in Base 9 — Upper bound on s
There is no (104, 125, large)-net in base 9, because
- 19 times m-reduction [i] would yield (104, 106, large)-net in base 9, but