Best Known (101, 122, s)-Nets in Base 9
(101, 122, 106288)-Net over F9 — Constructive and digital
Digital (101, 122, 106288)-net over F9, using
- net defined by OOA [i] based on linear OOA(9122, 106288, F9, 21, 21) (dual of [(106288, 21), 2231926, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9122, 1062881, F9, 21) (dual of [1062881, 1062759, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9122, 1062884, F9, 21) (dual of [1062884, 1062762, 22]-code), using
- trace code [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]
- trace code [i] based on linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9122, 1062884, F9, 21) (dual of [1062884, 1062762, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9122, 1062881, F9, 21) (dual of [1062881, 1062759, 22]-code), using
(101, 122, 1062888)-Net over F9 — Digital
Digital (101, 122, 1062888)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9122, 1062888, F9, 21) (dual of [1062888, 1062766, 22]-code), using
- trace code [i] based on linear OA(8161, 531444, F81, 21) (dual of [531444, 531383, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(8161, 531444, F81, 21) (dual of [531444, 531383, 22]-code), using
(101, 122, large)-Net in Base 9 — Upper bound on s
There is no (101, 122, large)-net in base 9, because
- 19 times m-reduction [i] would yield (101, 103, large)-net in base 9, but