MinT - Best Known (109−22, 109, s)-Nets in Base 9

Best Known (109−22, 109, s)-Nets in Base 9

(109−22, 109, 5388)-Net over F₉ — Constructive and digital

Digital (87, 109, 5388)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (74, 96, 5368)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9⁹⁶, 5368, F₉, 22, 22) (dual of [(5368, 22), 118000, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(9⁹⁶, 59048, F₉, 22) (dual of [59048, 58952, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(9⁹⁶, 59049, F₉, 22) (dual of [59049, 58953, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(109−22, 109, 97369)-Net over F₉ — Digital

Digital (87, 109, 97369)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−22, 109, large)-Net in Base 9 — Upper bound on s

There is no (87, 109, large)-net in base 9, because

20 times m-reduction [i] would yield (87, 89, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]