Best Known (86, 86+27, s)-Nets in Base 9
(86, 86+27, 1011)-Net over F9 — Constructive and digital
Digital (86, 113, 1011)-net over F9, using
- 91 times duplication [i] based on digital (85, 112, 1011)-net over F9, using
- net defined by OOA [i] based on linear OOA(9112, 1011, F9, 27, 27) (dual of [(1011, 27), 27185, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9112, 13144, F9, 27) (dual of [13144, 13032, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9112, 13146, F9, 27) (dual of [13146, 13034, 28]-code), using
- trace code [i] based on linear OA(8156, 6573, F81, 27) (dual of [6573, 6517, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- trace code [i] based on linear OA(8156, 6573, F81, 27) (dual of [6573, 6517, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9112, 13146, F9, 27) (dual of [13146, 13034, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9112, 13144, F9, 27) (dual of [13144, 13032, 28]-code), using
- net defined by OOA [i] based on linear OOA(9112, 1011, F9, 27, 27) (dual of [(1011, 27), 27185, 28]-NRT-code), using
(86, 86+27, 18526)-Net over F9 — Digital
Digital (86, 113, 18526)-net over F9, using
(86, 86+27, large)-Net in Base 9 — Upper bound on s
There is no (86, 113, large)-net in base 9, because
- 25 times m-reduction [i] would yield (86, 88, large)-net in base 9, but