Best Known (121, 121+25, s)-Nets in Base 9
(121, 121+25, 88573)-Net over F9 — Constructive and digital
Digital (121, 146, 88573)-net over F9, using
- net defined by OOA [i] based on linear OOA(9146, 88573, F9, 25, 25) (dual of [(88573, 25), 2214179, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9146, 1062877, F9, 25) (dual of [1062877, 1062731, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9146, 1062884, F9, 25) (dual of [1062884, 1062738, 26]-code), using
- trace code [i] based on linear OA(8173, 531442, F81, 25) (dual of [531442, 531369, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- trace code [i] based on linear OA(8173, 531442, F81, 25) (dual of [531442, 531369, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9146, 1062884, F9, 25) (dual of [1062884, 1062738, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9146, 1062877, F9, 25) (dual of [1062877, 1062731, 26]-code), using
(121, 121+25, 1062888)-Net over F9 — Digital
Digital (121, 146, 1062888)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9146, 1062888, F9, 25) (dual of [1062888, 1062742, 26]-code), using
- trace code [i] based on linear OA(8173, 531444, F81, 25) (dual of [531444, 531371, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(8173, 531441, F81, 25) (dual of [531441, 531368, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(8173, 531444, F81, 25) (dual of [531444, 531371, 26]-code), using
(121, 121+25, large)-Net in Base 9 — Upper bound on s
There is no (121, 146, large)-net in base 9, because
- 23 times m-reduction [i] would yield (121, 123, large)-net in base 9, but