MinT - Best Known (90−12, 90, s)-Nets in Base 9

Best Known (90−12, 90, s)-Nets in Base 9

(90−12, 90, 2796200)-Net over F₉ — Constructive and digital

Digital (78, 90, 2796200)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁹⁰, 2796200, F₉, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(9⁹⁰, 8388601, F₉, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
  - discarding factors / shortening the dual code based on linear OOA(9⁹⁰, 8388602, F₉, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
    - trace code [i] based on linear OOA(81⁴⁵, 4194301, F₈₁, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(81⁴⁵, 8388602, F₈₁, 12) (dual of [8388602, 8388557, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁴⁵, large, F₈₁, 12) (dual of [large, large−45, 13]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(90−12, 90, large)-Net over F₉ — Digital

Digital (78, 90, large)-net over F₉, using

9¹ times duplication [i] based on digital (77, 89, large)-net over F₉, using
- t-expansion [i] based on digital (76, 89, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁸⁹, large, F₉, 13) (dual of [large, large−89, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 9⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(90−12, 90, large)-Net in Base 9 — Upper bound on s

There is no (78, 90, large)-net in base 9, because

10 times m-reduction [i] would yield (78, 80, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]