Best Known (86, 111, s)-Nets in Base 9
(86, 111, 4921)-Net over F9 — Constructive and digital
Digital (86, 111, 4921)-net over F9, using
- net defined by OOA [i] based on linear OOA(9111, 4921, F9, 25, 25) (dual of [(4921, 25), 122914, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9111, 59053, F9, 25) (dual of [59053, 58942, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9111, 59054, F9, 25) (dual of [59054, 58943, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(9111, 59054, F9, 25) (dual of [59054, 58943, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9111, 59053, F9, 25) (dual of [59053, 58942, 26]-code), using
(86, 111, 43152)-Net over F9 — Digital
Digital (86, 111, 43152)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9111, 43152, F9, 25) (dual of [43152, 43041, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using
(86, 111, large)-Net in Base 9 — Upper bound on s
There is no (86, 111, large)-net in base 9, because
- 23 times m-reduction [i] would yield (86, 88, large)-net in base 9, but