Best Known (81, 98, s)-Nets in Base 9
(81, 98, 132860)-Net over F9 — Constructive and digital
Digital (81, 98, 132860)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 132860, F9, 17, 17) (dual of [(132860, 17), 2258522, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(998, 1062881, F9, 17) (dual of [1062881, 1062783, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(998, 1062884, F9, 17) (dual of [1062884, 1062786, 18]-code), using
- trace code [i] based on linear OA(8149, 531442, F81, 17) (dual of [531442, 531393, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- trace code [i] based on linear OA(8149, 531442, F81, 17) (dual of [531442, 531393, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(998, 1062884, F9, 17) (dual of [1062884, 1062786, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(998, 1062881, F9, 17) (dual of [1062881, 1062783, 18]-code), using
(81, 98, 1062888)-Net over F9 — Digital
Digital (81, 98, 1062888)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(998, 1062888, F9, 17) (dual of [1062888, 1062790, 18]-code), using
- trace code [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8149, 531444, F81, 17) (dual of [531444, 531395, 18]-code), using
(81, 98, large)-Net in Base 9 — Upper bound on s
There is no (81, 98, large)-net in base 9, because
- 15 times m-reduction [i] would yield (81, 83, large)-net in base 9, but