MinT - Best Known (89, 89+27, s)-Nets in Base 9

Best Known (89, 89+27, s)-Nets in Base 9

(89, 89+27, 1012)-Net over F₉ — Constructive and digital

Digital (89, 116, 1012)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹¹⁶, 1012, F₉, 27, 27) (dual of [(1012, 27), 27208, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9¹¹⁶, 13157, F₉, 27) (dual of [13157, 13041, 28]-code), using
  - discarding factors / shortening the dual code based on linear OA(9¹¹⁶, 13158, F₉, 27) (dual of [13158, 13042, 28]-code), using
    - trace code [i] based on linear OA(81⁵⁸, 6579, F₈₁, 27) (dual of [6579, 6521, 28]-code), using
      - constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [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(81⁴¹, 6562, F₈₁, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(81⁵, 17, F₈₁, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
        discarding factors / shortening the dual code based on linear OA(81⁵, 81, F₈₁, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
        Reedâ€“Solomon code RS(76,81) [i]

(89, 89+27, 23869)-Net over F₉ — Digital

Digital (89, 116, 23869)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+27, large)-Net in Base 9 — Upper bound on s

There is no (89, 116, large)-net in base 9, because

25 times m-reduction [i] would yield (89, 91, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]