MinT - Best Known (85, 112, s)-Nets in Base 9

Best Known (85, 112, s)-Nets in Base 9

(85, 112, 1011)-Net over F₉ — Constructive and digital

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
      - constructionÂ X applied to C([0,13]) âŠ‚ 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 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 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 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 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
        Reedâ€“Solomon code RS(78,81) [i]

(85, 112, 17026)-Net over F₉ — Digital

Digital (85, 112, 17026)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 112, large)-Net in Base 9 — Upper bound on s

There is no (85, 112, large)-net in base 9, because

25 times m-reduction [i] would yield (85, 87, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]